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The Ultimate Q51 Guide [Expert Level]

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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   12 Sep 2020, 01:05
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MathRevolution wrote:
(Number Properties) p, q, and r are positive integers. What is the value of p + q + r?

1) p, q, and r are prime numbers.
2) The product of p, q, and r is 5 times the sum of p, q, and r.



Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to determine the value of p + q + r.

Follow the second and the third step: From the original condition, we have 3 variables (p, q, and r). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Since we have 3 variables (p, q, and r) and 0 equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) together give us that:

Since we have pqr = 5(p + q + r), we assume p = 5.

Then we have 5qr = 5(5 + q + r) or qr = q + r + 5.

Since we have qr – q – r = 5, we have qr – q – r + 1 = 6 (by adding 1 to both sides so that it can easily be factored). Factoring gives us (q - 1)(r - 1) = 6.

Then we have the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2 and r = 7.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3 and r = 4.
If we have q – 1 = 3 and r – 1 = 2, we have q = 4 and r = 3.
If we have q – 1 = 6 and r – 1 = 1, we have q = 7 and r = 2.

However, p, q, and r are prime numbers from condition 1), and their possible numbers are 2, 5, and 7.

Thus, we have p + q + r = 2 + 5 + 7 = 14.

The answer is unique, so both conditions together are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one. So, C seems to be the answer.

However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition 1) tells us that we don’t have a unique solution, obviously.

If p = 2, q = 3, and r = 5, we have p + q + r = 2 + 3 + 5 = 10.
If p = 2, q = 5, and r = 7, we have p + q + r = 2 + 5 + 7 = 14.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that we don’t have a unique solution.

Assume p = 5 and since we have qr – q – r = 5, we have qr – q – r + 1 = 6 or (q - 1)(r - 1) = 6.

Then the possible pairs of q - 1 and r - 1 are (1, 6), (2, 3), (3, 2), and (6, 1).

If we have q – 1 = 1 and r – 1 = 6, we have q = 2, r = 7 and p + q + r = 5 + 2 + 7 = 14.
If we have q – 1 = 2 and r – 1 = 3, we have q = 3, r = 4 and p + q + r = 5 + 3 + 4 = 12.

The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Thus, really, both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   13 Sep 2020, 13:15
Exactly what I needed. Thanks for putting this together.
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   14 Sep 2020, 05:50
Expert Reply
MathRevolution wrote:
(Geometry) The figure shows a triangle ABC and points D, E, and F, which are on line AB, BC, and CA, respectively. The area of triangle ABC is 15. What is the area of triangle BCF?

Attachment:
Triangle.jpg
Triangle.jpg [ 7.03 KiB | Viewed 2606 times ]


1) (BE):(EC)=3:4
2) The area of △BCF is equal to the area of □ECFD.


Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the area of triangle BCF, and since the area of triangle ABC is 15, we should know the ratio of AF: FC to determine the area of triangle BCF.

Follow the second and the third step: From the original condition, we have many variables (many triangles and each triangle has 3 variables). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer.

Let’s look at both conditions 1) & 2) together.

Attachment:
Triangle -Solution-.jpg
Triangle -Solution-.jpg [ 7.69 KiB | Viewed 2549 times ]


Since we have △BCF = △BEF + △ECF, □ECFD = △DEF + △ECF and △BCF = □ECFD, we have △BEF = △DEF.

Since triangles BEF and DEF have a common base of EF, their heights are equal to each other. Thus, AB and EF are parallel to each other.

Then triangles CEF and CBA are similar.

Since CF:AC = EC:BC = 4:7, we have △BCF = (\(\frac{4}{7}\)) △ABC = (\(\frac{4}{7}\)) * 15 = \(\frac{60}{7}\).

The answer is unique, so both conditions are sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   17 Sep 2020, 04:43
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MathRevolution wrote:
(Geometry) What is the value of ∠x + ∠y?

Attachment:
DS Triangle .jpg
DS Triangle .jpg [ 9.27 KiB | Viewed 2501 times ]


1) ∠BAC = \(40^o\).
2) ∠ABD = ∠DBE = ∠EBC.



Solution:


Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of ∠x + ∠y.

Follow the second and the third step: From the original condition, we have many variables (many triangles and each triangle has 3 variables). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.
Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions 1) & 2) together.

If ∠ABD = ∠DBE = ∠EBC = \(20^o\), ∠x = \(40^o\), and ∠y = \(20^o\), we have ∠x + ∠y = \(60^o\)°.

If ∠ABD = ∠DBE = ∠EBC = \(15^o\), ∠x = \(30^o\), ∠y = \(15^o\), we have ∠x + ∠y = \(45^o\).

The answer is not unique, and the conditions combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D. 
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   19 Sep 2020, 10:34
Expert Reply
MathRevolution wrote:
Q1. (Numbers) N is a 3-digit positive integer. a is the hundreds digit, b the tens digit, and c the units digit. What is the maximum possible value of N?

1) b > 2a + c.
2) c > 0.



Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

Let N = 11a + 10b + c. Then we have to find the maximum possible value of N.

Since N is a three-digit integer, we need the value of a to be the maximum possible value, and b > c.

Follow the second and the third step: From the original condition. We have 4 variables (N, a, b, and c) and 1 equation (N = 100a + 10b + c). To match the number of variables with the number of equations, we need 3 more equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions (1) and (2) together. They tell us that b > 2a + c and c > 0.

The maximum value b can take is 9. For 9 > 2a + c, let’s find values for a and c.

=> c > 0 means the values c can have start at 1.

=> 9 > 2a + 1 – In this case a = 3.
=> 9 > 2a + 2 – In this case a = 3
=> 9 > 2a + 3 – In this case a = 2.

The largest possible value of a is 3, and the maximum value of c is then 2.

Then, N = 100*3+ 10*9 + 2 = 392. The answer is unique, and both conditions (1) and (2) combined are sufficient, according to CMT 2, which states that the number of answers must be only one. So, C seems to be the answer.

Since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition (1) tells us that b > 2a + c, from which we get that for b to be greater than (2a + c), its value depends on a and c. We have digits from 0 to 9, so b cannot be greater than 9. Suppose b = 9. That means a can be 3, and c can be 2. Then we get 2a + c = 2*3 + 2 = 8 since 9 is greater than 8.

However, if a = 0, then b > c and c can have any value from 0 to 8.

Similarly, if c is 0, then b > 2a and a can have values from 0 to 4.

The answer is not unique, and the condition is not sufficient, according to CMT 2, which states that the number of answers must be only one.

Condition (2) tells us that c > 0, from which we cannot determine anything about the value of a.

The answer is not unique, and the condition is not sufficient, according to CMT 2, which states that the number of answers must be only one.

So, really, both conditions (1) and (2) combined are sufficient.

Both conditions (1) and (2) together are sufficient.

Therefore, C is the correct answer.

Answer: C
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   26 Sep 2020, 08:17
Expert Reply
MathRevolution wrote:
Q. (Number) What is the value of a positive integer n?

1) 756n + 576 is a perfect square number.
2) n is a unit digit number.



Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of 'n'.

Follow the second and the third step: From the original condition, we have 1 variable (n). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3 Principles and choose D as the most likely answer.

Let’s look at each condition separately.

Condition (1) tells us that 756n + 576 is a perfect square number , from which we get as below,

756n + 576 = (2*2*3*3*3*7)n + 2*2*2*2*2*2*3*3
= (36*21)n + 36*16 = 36(21n + 16)

Since 756n + 576 is a perfect square, (21n + 16) has to be a perfect square as 36 is already the perfect square of 6. We can find the value of n as shown below:

=> 21n + 16 = 36 => 21n = 20 => n = not an integer.
=> 21n + 16 = 49 => 21n = 33 => n = not an integer.
=> 21n + 16 = 64 => 21n = 48 => n = not an integer.
=> 21n + 16 = 81 => 21n = 65 => n = not an integer.
21n + 16 = 100 => 21n = 84 => n = 4.
21n + 16 = 121 => 21n = 105 => n = 5.

So, the values 4 or 5 are possible for n.

The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.


Condition (2) tells us that n is a unit digit number, from which it follows that n can take the values from 0 to 9.

The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.

Let’s look at both conditions together. They tell us that the values of n can be 4 or 5.

The answer is not unique, so both conditions (1) and (2) combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.

Both conditions (1) and (2) together are not sufficient.

Therefore, E is the correct answer.

Answer: E
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   02 Oct 2020, 07:33
Expert Reply
MathRevolution wrote:
Q. (Number) If x and y are positive integers, what is the value of \(\frac{x}{(x + y)}\)?

1) \(\frac{y}{x}\) = \(\frac{(y - 39)}{(x - 21)}\).

2) The least common multiple of x and y is 1001.



Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of\( \frac{x}{(x + y)}\) for positive integers x and y.

Follow the second and the third step: From the original condition, we have 2 variables (x and y). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.

Recall 3 Principles and choose C as the most likely answer.

Let’s look at both conditions (1) and (2) together.

Conditions (1) and (2) tell us that \(\frac{y}{x}\) = \(\frac{(y - 39)}{(x - 21)}\) and that the least common multiple of x and y is 1001. Then, we get y(x - 21) = x(y - 39), xy - 21y = xy - 39x.

=> If we subtract xy from both sides, we get, -21y = -39x, and dividing both sides by -1 gives us 21y = 39x.

=> If we divide both sides by 3, we get 7y = 13x or the ratio of x : y = 7 : 13.

=> We can have multiple combinations that will give us the ratio of 7:13 between x and y.

=> We know that 1001 is divisible by both x and y from condition (2). Since 1001 = 7*11*13, x should be 7*11 and y should be 13*11, so we get \(\frac{x}{(x + y)}\) = \(\frac{(7*11 )}{(7*11 + 13*11 )}\) = \(\frac{(7)}{(7 + 13)}\) = \(\frac{(7 )}{(20)}\).

The answer is unique, so both conditions (1) and (2) combined are sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one. So, C seems to be the answer.

Since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer.

Let’s look at each condition separately:

Condition (1) tells us that x : y = 7 : 13 as shown above. To form a ratio, we need a common number. Let the common number be k. So, we get x = 7k and y = 13k.

Then, \(\frac{x}{(x + y)}\) = \(\frac{(7k )}{(7k + 13k )}\) = \(\frac{(7k )}{(20k )}\) = \(\frac{(7)}{(20)}\).

The answer is unique, so the condition is sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.

Condition (2) tells us that the least common multiple of x and y is 1001. Then. since 1001 = 7*11*13, (x, y) can be (7*11, 13*11), (7*11, 7*13), or (7*13, 13*11).

The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.

If the question has both C and A as its answer, then A is the answer rather than C according to the definition of DS questions.

Here, we should know that condition (1) with a ratio wins over condition (2) with a number.

Also, we should remember the solving process of the relationship between the Variable Approach, and Common Mistake Type 3, and Common Mistake Type 4(A or B).

Also, we should know CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer.


Condition (1) alone is sufficient.

Therefore, A is the correct answer.

Answer: A




In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   04 Oct 2020, 13:18
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MathRevolution wrote:
How many digits does \(2^{17}*5^{10}\) have?


A. 11
B. 13
C. 15
D. 17
E. 19



Solution:

To find the number of digits, pair up 2 and 5 with common powers, and we get

=> \(2^{17}\) * \(5^{10}\) = \(2^{7}\) * \(2^{10}\) * \(5^{10}\)

=> If we use the exponent property of \(a^x\) * \(b^x\) = \((ab)^x\), we get

=>\(2^{17}\) * \(5^{10}\) = \(2^{7}\) * \(2^{10}\) * \(5^{10}\) = 128 * \(10^{10}\)

=> Since 128 * \(10^{10}\) has 128 as the first 3 digits followed by 10 zeros, the total number of digits of \(2^{17}\) * \(5^{10}\) = 3 + 10 = 13.

Therefore, B is the correct answer.

Answer B
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   07 Oct 2020, 12:28
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Solution:

Divisors or factors of a number, say M, are the integers that can divide into M without a remainder. That is, if we have a number M, such that M = ab and a and b are positive integers, then a and b are factors of M.

When M is expressed as a product of its prime factors only, then we say that we have prime factorized M. If we prime factorize a positive integer, M, as M = \({p_1}^{t_1} \) *\({p_2}^{t_2} \) * ……*\({p_n}^{t_n} \), where \(p_i\) stands for different prime numbers, and \(t_i\) are positive integers and stands for the exponents of the different prime factors or divisors, then the number of factors of M = (\(t_1\) + 1)·(\(t_2\) + 1)......(\(t_n\) + 1).

The important part here is the word “different.”

In the question, n(A) denotes the number of positive divisors of a natural number A. We are required to find the total number of A’s that satisfy n(A) = 3 between 1 and 50, inclusive.

To have 3 positive divisors, A must have a single prime factor with the highest power of 2. This is possible when prime numbers are squared.

=> \(2^2\) = 4 → 3 divisors → 1, 2, and 4.

=> \(3^2\) = 9 → 3 divisors → 1, 3, and 9.

=> \(5^2\) = 25 → 3 divisors → 1, 5, and 25.

=> \(7^2\) = 49 → 3 divisors → 1, 7, and 49.



Hence, there are 4 A’s that satisfy n(A) = 3 between 1 and 50, inclusive.

Therefore, A is the correct answer

Answer A
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   11 Oct 2020, 11:58
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MathRevolution wrote:
(Number) What are the values of positive integers p, q, and r?

1) p, q, and r are prime numbers and p < q < r.

2) The product of p, q, and r is 5 times the sum of p, q, and r.



Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the values of natural numbers p, q, and r.

Follow the second and third steps: From the original condition, we have 3 variables (p, q, and r). To match the number of variables with the number of equations, we need 3 more equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer

Let's take a look at both conditions (1) and (2) together. Then we get:

=> p < q < r and pqr = 5 * (p + q + r)

=> p = 2 , q = 5 and r = 7

=> pqr = 70 and p + q + r = 14

Since the answer is unique, both conditions (1) and (2) combined are sufficient, according to CMT 2, which states that the number of answers must be one. So, C seems to be the answer.

However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition (1) tells us that p, q, and r are prime numbers and p < q < r.

=> If p = 2, q = 3, and r = 5 then p < q < r.

=> If p = 3, q = 5, and r = 7 then p < q < r.

Since the answer is not unique, the condition is not sufficient, according to CMT 2, which states that the number of answers must be one.

Condition (2) tells us that the product of p, q, and r is 5 times the sum of p, q, and r.

=> Thus, pqr = 5 (p + q + r). This means that pqr is a multiple of 5, and thus one of the prime numbers is 5.

=> If p = 5 , q = 7, and r = 2 then pqr = 5 * 7 * 2 = 70 and p + q + r = 5 + 7 + 2 = 14. Therefore, pqr = 5 (p + q + r).

However, if p = 2, q = 7, and r = 5 then pqr = 2 * 7 * 5 = 70 and p + q + r = 2 + 7 + 5 = 14. Therefore, pqr = 5 (p + q + r).

Since the answer is not unique, the condition is not sufficient, according to CMT 2, which states that the number of answers must be one.

So, both conditions (1) and (2) together are sufficient.

Therefore, C is the correct answer.

Answer C
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   17 Oct 2020, 01:15
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MathRevolution wrote:
[GMAT math practice question]

(Number Properties) \(a\) and \(b\) are positive integers. What is the value of \(2^a + 2^b\)?

1) \(a\) is the units digit of \(7^{1020}\) and \(b\) is the units digit of \(3^{224}.\)
2) \(a\) and \(b\) are neither prime numbers nor composite numbers.


=>

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Since we have \(2\) variables (\(a\) and \(b\)) and \(0\) equations and each condition has 2 equations, C is most likely to be the answer. Let’s look at both conditions together. However, since the value of condition (1) is equal to the value of condition (2), by Tip 1, we get D as the most likely answer. Let’s look at each condition separately.

Let’s look at the condition 1). It tells us that \(a = 1\) and \(b = 1\).

Units of powers of \(7\) are \(7^1\)~\(7\), \(7^2\)~\(9\), \(7^3\)~\(3\), \(7^4\)~\(1\), \(7^5\)~\(7\), …

So, the units digits of \(7^n\) have a period of \(4\):

They form the cycle \(7 -> 9 -> 3 -> 1\).

Thus, \(7^n\) has a units digit of \(1\) if \(n\) has a remainder of \(0\) when it is divided by \(4\).

The remainder is \(0\) when \(224\) is divided by \(4\), so the units digit of \(7^{1020}\) is \(1\).

Units of powers of \(3\) are \(3^1\)~\(3\), \(3^2\)~\(9\), \(3^3\)~\(7\), \(3^4\)~\(1\), \(3^5\)~\(3\), …

So, the units digits of \(3^n\) have a period of \(4:\)

They form the cycle \(3 -> 9 -> 7 -> 1.\)

Thus, \(3^n\) has a units digit of \(1\) if \(n\) has a remainder of \(0\) when it is divided by \(4\).

The remainder is \(0\) when \(224\) is divided by \(4\), so the units digit of \(3^{224}\) is 1.

\(2^1 + 2^1 = 2 + 2 = 4.\)

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Let’s look at the condition 2), it tells us that \(a = 1\) and \(b = 1.\)

\(2^1 + 2^1 = 2 + 2 = 4.\)

The answer is unique, so the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Also, the original condition needs 2 equations.
Condition 1) has 2 equations.
Condition 2) has 2 equations.


Each condition ALONE is sufficient.

Therefore, D is the correct answer.
Answer: D

Note Tip 1) of the VA method states that D is most likely the answer if condition 1) gives the same information as to condition 2).

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations,” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   19 Oct 2020, 11:16
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MathRevolution wrote:
[GMAT math practice question]

(Statistics) 100 students take a test. 20 students are in class A, 30 students in class B, and 50 students in class C. What is the average of the 100 students?

1) The average of class B is 10 points higher than that of class A.
2) The average of class C is 20 points higher than that of class B.


=>

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Visit https://www.mathrevolution.com/gmat/lesson for details.

Assume \(a, b\) and \(c\) are the averages of classes \(A, B\), and \(C\), respectively.

Since we have \(3\) variables (\(a, b\), and \(c\)) and \(0\) equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) together give us that \(b = a + 10\) and \(c = b + 20:\)

If \(a = 60, b = 70\) and \(c = 90\), then the average is
\(\frac{60·20 + 70·30 + 90·50}{100} = \frac{1200 + 2100 + 4500}{100 }= \frac{7800}{100} = 78\)

If \(a = 50, b = 60\) and \(c = 80\), then the average is
\(\frac{50·20 + 60·30 + 80·50}{100} = \frac{1000 + 1800 + 4000}{100} = \frac{6800}{100} = 68\)

The answer is not unique, and both conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are not sufficient.

Therefore, E is the correct answer.
Answer: E

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   21 Oct 2020, 01:26
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[GMAT math practice question]

(Algebra) What is the value of \((a-b)^2\)?

1) \(\frac{b}{a} < 0\).

2) \(|a| = 4\) and \(|b| = 3\).


Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 2 variables (a and b) and 0 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) together give us that \(a = ±4, b = ±3\) and \(ab < 0\).
If \(a = 4\) and \(b = -3\), then \((4-(-3))^2 = 7^2 = 49.\)
If \(a = -4\) and \(b = 3\), then \((-4-3)^2 = 7^2 = 49.\)

The answer is unique, yes, so both conditions are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Both conditions 1) & 2) together are sufficient.

Therefore, C is the correct answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in Common Mistake Types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   05 Nov 2020, 02:32
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[GMAT math practice question]

<\(x\)> denotes the unit digit of \(x\). For example, <\(123\)> = \(3\). What is <\(7^{19}+7^{89}\)>?

A. \(0\)

B. \(1\)

C. \(3\)

D. \(8\)

E. \(9\)


Solution:

The unit digit of the powers of 7 repeats 7->9->3->1 and has a period of 4.

Since 19 = 4·4 + 3 and 19 has a remainder 3 when it is divided by 4, the units digit of 719 is 3, 719 ~ 73 ~ 3.

Since 89 = 4·22 + 1 and 89 has a remainder 1 when it is divided by 4, the units digit of 789 is 7, 789 ~ 71 ~ 7.

< 719 + 719 > ~ < 3 + 7 > ~ <10> ~ 0.

Therefore, A is the correct answer.

Answer: A
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   10 Nov 2020, 10:41
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MathRevolution wrote:
Q. (Number) What is the value of a positive integer n?

1) 756n + 576 is a perfect square number.
2) n is a unit digit number.



Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

We have to find the value of 'n'.

Follow the second and the third step: From the original condition, we have 1 variable (n). To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Recall 3 Principles and choose D as the most likely answer.

Let’s look at each condition separately.

Condition (1) tells us that 756n + 576 is a perfect square number , from which we get as below,

756n + 576 = (2*2*3*3*3*7)n + 2*2*2*2*2*2*3*3
= (36*21)n + 36*16 = 36(21n + 16)

Since 756n + 576 is a perfect square, (21n + 16) has to be a perfect square as 36 is already the perfect square of 6. We can find the value of n as shown below:

=> 21n + 16 = 36 => 21n = 20 => n = not an integer.
=> 21n + 16 = 49 => 21n = 33 => n = not an integer.
=> 21n + 16 = 64 => 21n = 48 => n = not an integer.
=> 21n + 16 = 81 => 21n = 65 => n = not an integer.
21n + 16 = 100 => 21n = 84 => n = 4.
21n + 16 = 121 => 21n = 105 => n = 5.

So, the values 4 or 5 are possible for n.

The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.


Condition (2) tells us that n is a unit digit number, from which it follows that n can take the values from 0 to 9.

The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.

Let’s look at both conditions together. They tell us that the values of n can be 4 or 5.

The answer is not unique, so both conditions (1) and (2) combined are not sufficient, according to Common Mistake Type 2, which states that the number of answers should be only one.

Both conditions (1) and (2) together are not sufficient.

Therefore, E is the correct answer.

Answer: E
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   02 Dec 2020, 12:33
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MathRevolution wrote:
n(A) denotes the number of positive divisors of a positive integer A. What is the smallest possible value of x satisfying \(\frac{n(280)·n(x)}{ n(30)}\) = 12?

A. 12
B. 14
C. 16
D. 18
E. 20



Solution:


If n = \(p^a\)·\(q^b\)·\(r^c\) has a prime factorization with different prime numbers p, q, and r, then the number of factors of n is (a + 1)(b + 1)(c + 1).

Since 280 = \(2^3\)·\(5^1\)·\(7^1\), we have n(280) = (3 + 1)(1 + 1)(1 + 1) = 4·2·2 = 16.

Since 30 = \(2^1\)·\(3^1\)·\(5^1\), we have n(30) = (1 + 1)(1 + 1)(1 + 1) = 2·2·2 = 8.

Then we have \(\frac{n(280) * n(x)}{n(30)}\) = \(\frac{16n(x)}{8}\) = 2n(x) = 12 or n(x) = 6.

We have two cases of integers with 6 factors. They are \(p^2\)\(q^1\) or \(p^5\) where p and q are different prime numbers since (2 + 1)(1 + 1) = 6 and (5 + 1) = 6.

For x = \(p^2\)\(q^1\), when we have p = 2 and q = 3, we have the smallest number \(2^2\)·\(3^1\) = 12.

For x = \(p^5\), when we have p = 2, we have the smallest number \(2^5\) = 32.

The smallest value of x is 12.

Therefore, A is the correct answer.

Answer: A
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   03 Dec 2020, 03:46
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MathRevolution wrote:
(Functions): If 2f (x) + 3f (−x) = 2x − 4, what is the value of 5f (1)?

A) 14/5
B) -2
C) 14
D) -14/5
E) -14



Thank you for your replies GMAT Club members. GMAT quant is based on logic, tricks, and quick approaches. Always try to find a quick approach to solve any PS or a DS question. We apply the IVY approach for PS and Variable Approach for DS.


Solution: 2f(x) + 3 f(-x) = 2x - 4 --------- equation (1)

Substituting x = 1 in equation (1)

2f(1) + 3f(-1) = 2(1) – 4 =2 – 4= -2

2f(1) + 3f(-1) = -2--------- equation (2)

Substituting x = -1 in equation (1)

2f(-1) + 3f(1) = 2(-1) – 4 = -2 – 4 = -6

2f(-1) + 3f(1) = -6--------- equation (3)

Multiplying equation (2) by ‘2’ we get,

4f(1) + 6f(-1) = -4--------- equation (4)


Multiplying equation (3) by ‘3’ we get,

9f(1) + 6f(-1) = -18--------- equation (5)

Subtracting Eqn(5) - Eqn(4)

=> 5f(1) = -14

=> f(1) = \(\frac{-14}{5}\)

D is the correct answer.

Answer D
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   04 Dec 2020, 04:52
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MathRevolution wrote:
Q1. (Numbers) N is a 3-digit positive integer. a is the hundreds digit, b the tens digit, and c the units digit. What is the maximum possible value of N?

1) b > 2a + c.
2) c > 0.



Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Visit https://www.mathrevolution.com/gmat/lesson for details.

Now we will solve this DS question using the Variable Approach.

Let’s apply the 3 steps suggested previously.

Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.

Let N = 11a + 10b + c. Then we have to find the maximum possible value of N.

Since N is a three-digit integer, we need the value of a to be the maximum possible value, and b > c.

Follow the second and the third step: From the original condition. We have 4 variables (N, a, b, and c) and 1 equation (N = 100a + 10b + c). To match the number of variables with the number of equations, we need 3 more equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Recall 3 Principles and choose E as the most likely answer. Let’s look at both conditions (1) and (2) together. They tell us that b > 2a + c and c > 0.

The maximum value b can take is 9. For 9 > 2a + c, let’s find values for a and c.

=> c > 0 means the values c can have start at 1.

=> 9 > 2a + 1 – In this case a = 3.
=> 9 > 2a + 2 – In this case a = 3
=> 9 > 2a + 3 – In this case a = 2.

The largest possible value of a is 3, and the maximum value of c is then 2.

Then, N = 100*3+ 10*9 + 2 = 392. The answer is unique, and both conditions (1) and (2) combined are sufficient, according to CMT 2, which states that the number of answers must be only one. So, C seems to be the answer.

Since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which states that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.

Condition (1) tells us that b > 2a + c, from which we get that for b to be greater than (2a + c), its value depends on a and c. We have digits from 0 to 9, so b cannot be greater than 9. Suppose b = 9. That means a can be 3, and c can be 2. Then we get 2a + c = 2*3 + 2 = 8 since 9 is greater than 8.

However, if a = 0, then b > c and c can have any value from 0 to 8.

Similarly, if c is 0, then b > 2a and a can have values from 0 to 4.

The answer is not unique, and the condition is not sufficient, according to CMT 2, which states that the number of answers must be only one.

Condition (2) tells us that c > 0, from which we cannot determine anything about the value of a.

The answer is not unique, and the condition is not sufficient, according to CMT 2, which states that the number of answers must be only one.

So, really, both conditions (1) and (2) combined are sufficient.

Both conditions (1) and (2) together are sufficient.

Therefore, C is the correct answer.

Answer: C
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   05 Dec 2020, 09:58
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MathRevolution wrote:
How many digits does \(2^{17}*5^{10}\) have?


A. 11
B. 13
C. 15
D. 17
E. 19



Solution:

To find the number of digits, pair up 2 and 5 with common powers, and we get

=> \(2^{17}\) * \(5^{10}\) = \(2^{7}\) * \(2^{10}\) * \(5^{10}\)

=> If we use the exponent property of \(a^x\) * \(b^x\) = \((ab)^x\), we get

=>\(2^{17}\) * \(5^{10}\) = \(2^{7}\) * \(2^{10}\) * \(5^{10}\) = 128 * \(10^{10}\)

=> Since 128 * \(10^{10}\) has 128 as the first 3 digits followed by 10 zeros, the total number of digits of \(2^{17}\) * \(5^{10}\) = 3 + 10 = 13.

Therefore, B is the correct answer.

Answer B
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Re: The Ultimate Q51 Guide [Expert Level] [#permalink] The Ultimate Q51 Guide [Expert Level]   07 Dec 2020, 04:13
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MathRevolution wrote:
(Integers) If m, n, p, and q are distinct positive integers, greater than 1 such that mnpq = 660 and m<n<p<q, how many possible combinations of values exist for m, n, p, and q?

A) Two
B) Three
C) Four
D) Five
E) Seven


Thank you for your replies GMAT Club members. GMAT quant is based on logic, tricks, and quick approaches. Always try to find a quick approach to solve any PS or a DS question. We apply the IVY approach for PS and Variable Approach for DS.


Solution: Let us find the prime factors of 660.

660 can be written as 660 = 2 * 2 * 3 * 5 * 11.

We have an extra ‘2’ and this can be combined with other factors to generate different values.

Also, considering all other factors than ‘2’, we may combine to generate different values for m, n, p, and q.

Attachment:
Possible Combinations.jpg


Therefore, we have ‘4’ possible combinations.

C is the correct answer.

Answer C.
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