The Ultimate Q51 Guide [Expert Level]

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

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.



Therefore, we have ‘4’ possible combinations.

C is the correct answer.

Answer C.

=>

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.

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 \(2\) 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 \(x\) and \(y\) are positive integers and \(\frac{1}{x}+\frac{1}{y}=\frac{1}{5}\), from which we get \(\frac{1}{x}+\frac{1}{y}=\frac{1}{5}, \frac{y}{xy} + \frac{x}{xy} = \frac{1}{5}, \frac{y + x}{xy}=\frac{1}{5}, 5y + 5x = xy, xy – 5x – 5y = 0\), or \(xy – 5x – 5y + 25 = 25.\) We can factor \(xy – 5x – 5y + 25 = 25\) as follows: \((xy – 5x) + (-5y + 25) = 25, x(y – 5) + (-5)(y – 5) = 25\), which is equal to \((x - 5)(y - 5) = 25.\) We have \(3\) possible cases: \(x – 5 = 25\), and \(y – 5 = 1 / x – 5 = 5\), and \(y – 5 = 5 / x – 5 = 1\), and \(y – 5 = 25,\) since \(x\) and \(y\) are positive integers from condition (1).

Thus, the possible pairs of \(x\) and \(y\) are \(x = 6\), and \(y = \frac{30 }{ x} = 10\), and \(y = 10\) / and \(x = 30\), and \(y = 6.\)

If \(x = 6\) and \(y = 30\), then \(x + y = 36.\)

If \(x = 10\) and \(y = 10\), then \(x + y = 20.\)

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

Both conditions (1) and (2) together are not sufficient.
Therefore, E is the correct answer.
Answer: E

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.

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 f(2) + f(3) + f(5).

Follow the second and the third step: From the original condition, we have many variables to determine a function f(x). 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.

Since f(1) = 1, we have f(2) = f(1+1) = f(1) + f(1) + 1·1 = 1 + 1 + 1 = 3 using condition 2).
Then we have f(3) = f(2+1) = f(2) + f(1) + 2·1 = 3 + 1 + 2 = 6.

f(5) = f(3+2) = f(3) + f(2) + 3·2 = 6 + 3 + 6 = 15.

Thus, we have f(2) + f(3) + f(5) = 3 + 6 + 15 = 24.

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.

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.

=>

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify the conditions, if necessary.

Condition 1)
Since \(x > y\) and the average of \(x\) and \(y\) is \(9\), we have \(x > 9 > y.\)

Thus, the median of \(x, 9, 9\) and \(y\) is \(9\).

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)
Since \(\frac{( x + y + 18 )}{3} = 12\) or \(x + y + 18 = 36\), the average of \(x\) and \(y\) is \(9.\)

Condition 2) is sufficient by the same reasoning as condition 1).

Therefore, D is the answer.
Answer: D

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

This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.

=>

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 \(1\) variable (\(x\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.

Let’s look at the condition 1). It tells us just the definition of a function \(f(x)\), therefore not giving us enough information to solve the question.
The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 1, which states that the number of answers must be only one.

Let’s look at the condition 2). It tells us that \(f(420)·f(x) = 96\). Since we do not have a definition of \(f(x)\), the answer is not unique, and condition 2) is not sufficient according to Common Mistake Type 1, which states that the number of answers must be only one.

Both conditions 1) & 2) together tell us that \(x\) has \(4\) factors for the following reason.

Remember the property that if \(n = p^aq^br^c\) where \(p, q\), and \(r\) are different prime numbers, \(n\) has \((a + 1)(b + 1)(c + 1)\) factors.

Since we have \(420 = 2^23^15^17^1\), it has \((2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3·2·2·2 = 24\) factors.

We have \(f(420)·f(x) = 24·f(x) = 96\) or \(f(x) = 4\), which means \(x\) has \(4\) factors.

Then we have two possibilities for \(x\), which are \(x = p^3\) or \(x = p·q\) where \(p\) and \(q\) are different prime numbers.

Then we have \(2^3 = 8\) or \(2·3 = 6\) as the possible values of \(x\).

Therefore, the minimum is \(6\).

The answer is unique, and both conditions 1) and 2) together 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

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.

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

=>
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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
We have to find the value of \(a + b\).

Follow the second and the third step: From the original condition, we have \(2\) variables (\(a\) and \(b\)). 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 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.
Condition 1) tells us that \(a = -2\) and \(b = 2\). In order for the equation to have more than one solution, the corresponding coefficients on both sides must be equal, respectively. Then we have the left-hand side \(2(x + a) = 2x + 2a\) and we have \(2x + 2a = bx – 4.\) Since the equation \(2x + 2a = bx – 4\) has more than one solution, we have \(2 = b\) and \(2a = -4.\) Thus, condition 1) tells us that \(a = -2\) and \(b = 2.\)

Then we have \(a + b = -2 + 2 = 0.\)

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.
Condition 2) tells us that \(a = -b.\) Since we have \(|a| = |b|\) and \(ab < 0, a\) and \(b\) have different signs and \(a = -b\). Thus, we have \(a + b = 0. \)

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.
Each condition alone is sufficient.
Therefore, D is the correct answer.
Answer: D

Normally, in problems that 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.

=>

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 \(3\) variables (\(x, y,\) and \(z\)) 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 \(y = x + 1, z = x + 2\) and \(12 + 13 + 14 + 15 = x + x + 1 + x + 2\), which is equivalent to \(54 = 3x + 3, 3x = 51\), or \(x = 27.\)

Since \(x, y\), and \(z\) with \(x < y < z\) are consecutive integers, we have \(y = x + 1\) and \(z = x + 2.\)

When we replace \(y\) and \(z\) with \(x + 1\) and \(x + 2\) in the equation \(12 + 13 + 14 + 15 = x + y + z,\) we have \(54 = 3x + 3\) or \(3x = 51.\)
Thus, we have \(x = 17.\)

The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, C is the answer.
Answer: C

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.

=>

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 many variables to determine a function, 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)

\(f(13)=\frac{f(10) - 1}{f(10) + 1}=\frac{11 - 1}{11 + 1}=\frac{5}{6}.\)

\(f(16)=\frac{f(13) - 1}{f(13) + 1}=\frac{5}{6} - 1/\frac{5}{6} + 1=\frac{-1}{11}.\)

\(f(19)=\frac{f(16)-1}{f(16)+1}=\frac{-1}{11}-1/\frac{-1}{11}+1=\frac{-6}{5}\)

\(f(22)=\frac{f(16) - 1}{f(16) + 1}=\frac{-6}{5} - 1/\frac{-6}{5} + 1=11.\)

Since we have \(2008 = 4*5002 + 0\) has a remainder \(0\) when it is divided by \(4\),\( f(2008) = f(16) = \frac{-1}{11}.\)
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

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.

=>

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The question \((x - a)^2 = (x - b)^2\) is equivalent to \(x = \frac{(a + b)}{2}\) for the following reason.
\((x - a)^2 = (x - b)^2\)

⇔ \((x - a)^2 - (x - b)^2 = 0\)

⇔ \(((x - a) - (x - b)) ((x - a) + (x - b)) = 0\)

⇔ \((-a + b)(2x - (a + b)) = 0\)

⇔ \((2x - (a + b)) = 0\) (by dividing both sides by \(-a + b\) since \(a ≠ b\))

⇔ \(2x = (a + b)\) (by adding (\(a + b\)) to both sides)

⇔ \(x = \frac{(a + b)}{2}\) (by dividing both sides by \(2\))

So, we have to find the value of \(a + b\).

Thus, look at condition (2). It tells us that \(a + b = 7\), which is exactly what we are looking for. The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Condition (1) tells us that \(a – b = 3\), from which we cannot determine the unique value of \(a + b. \) 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.
Condition (2) ALONE is sufficient.
Therefore, B is the answer.
Answer: B

=>

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 many variables to determine a function, 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)

\(f(13)=\frac{f(10) - 1}{f(10) + 1}=\frac{11 - 1}{11 + 1}=\frac{5}{6}.\)

\(f(16)=\frac{f(13) - 1}{f(13) + 1}=\frac{5}{6} - 1/\frac{5}{6} + 1=\frac{-1}{11}.\)

\(f(19)=\frac{f(16)-1}{f(16)+1}=\frac{-1}{11}-1/\frac{-1}{11}+1=\frac{-6}{5}\)

\(f(22)=\frac{f(16) - 1}{f(16) + 1}=\frac{-6}{5} - 1/\frac{-6}{5} + 1=11.\)

Since we have \(2008 = 4*5002 + 0\) has a remainder \(0\) when it is divided by \(4\),\( f(2008) = f(16) = \frac{-1}{11}.\)
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C

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.

=>
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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
We have to find the value of \(a + b\).

Follow the second and the third step: From the original condition, we have \(2\) variables (\(a\) and \(b\)). 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 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.
Condition 1) tells us that \(a = -2\) and \(b = 2\). In order for the equation to have more than one solution, the corresponding coefficients on both sides must be equal, respectively. Then we have the left-hand side \(2(x + a) = 2x + 2a\) and we have \(2x + 2a = bx – 4.\) Since the equation \(2x + 2a = bx – 4\) has more than one solution, we have \(2 = b\) and \(2a = -4.\) Thus, condition 1) tells us that \(a = -2\) and \(b = 2.\)

Then we have \(a + b = -2 + 2 = 0.\)

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.
Condition 2) tells us that \(a = -b.\) Since we have \(|a| = |b|\) and \(ab < 0, a\) and \(b\) have different signs and \(a = -b\). Thus, we have \(a + b = 0. \)

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.
Each condition alone is sufficient.
Therefore, D is the correct answer.
Answer: D

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.

=>

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.

Since we have \(3\) variables (\(x, y\) and \(z\)) and \(0\) equations, E is most likely to be 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)
By condition 1), \(x < y\) since \(x < x + 1 ≤ | x + 1 | < y\).
By condition 2), \(y < z\) since \(y < z – 1 < z.\)
Therefore, \(x < y < z\).
Thus, both conditions 1) & 2) together are sufficient.

Since this question is an absolute value question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If \(x = 1, y = 3\), and \(z = 5,\) then the answer is ‘yes’.
If \(x = 1, y = 3,\) and \(z = 2.9\), then the answer is ‘no’ since z < y.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
If \(x = 1, y = 3,\) and \(z = 5\), then the answer is ‘yes’.
If \(x = 3.1, y = 3,\) and \(z = 5\), then the answer is ‘no’ since \(x > y\).
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Therefore, C is the answer.
Answer: C

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.

=>

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.

Since we have \(2\) variables (\(x\) and \(y\)) and \(0\) equations, C is most likely to be 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):
Since \(y = 7 – x, x^2+y^2=25\) is equivalent to \(x^2+(7-x)^2=25\) or \(2x^2 – 14x + 24 = 0.\)
This factors as \(2(x^2 – 7x + 12) = 0\) or \(2(x-3)(x-4) = 0.\)
So, \(x = 3\) and \(y = 4\), or \(x = 4\) and \(y = 3\).
Since \(y ≠3\), we must have \(x = 3\) and \(x\) can’t be \(4\).
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, both conditions are sufficient, when used together.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since \(y = 7 – x ≠ 3, x\) can’t be \(4\). So, we have a unique answer, which is “no”.

Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)

If \(x = 4\) and \(y = -3\), then \(x^2+y2 = 25\), and the answer is “yes”.
If \(x = 3\) and \(y = 4,\) then \(x^2+y2 = 25\), and the answer is “no”.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Therefore, A is the answer.
Answer: A

Normally, in problems that 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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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.
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Condition 1)

Since we have \(a = 1, b = 1,\) and \(c = 2\), we have

\(2(a^4 + b^4 + c^4) = 2(1^4 + 1^4 + 2^4) \)

\(= 2(1 + 1 + 16) \)

\(= 2*18 \)

\(= 36.\)

\(2(a^4 + b^4 + c^4) = 36\) is a perfect square and the answer is ‘yes’.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

Since \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca), \)

we have \(a^2 + b^2 + c^2 = -2(ab + bc + ca).\)

Then, rearranging the second formula gives us:
\((a^2 + b^2 + c^2)^2 = (-2(ab + bc + ca))^2 \)

\(= 4(ab + bc + ca)(ab + bc + ca)\)

\(= 4(a^2b^2 + ab^2c + a^2bc + ab^2c + b^2c^2 + abc^2 + a^2bc + abc^2 + a^2c^2)\)

\(= 4((a^2b^2 + b^2c^2 + a^2c^2 + 2ab^2c + 2a^2bc + 2abc^2) \)

\(= 4(a^2b^2 + b^2c^2 + c^2a^2 + 2abc(a + b + c))\)

\(= 4(a^2b^2 + b^2c^2 + c^2a^2), \)since \(a + b + c = 0\)

Following the pattern in the first equation gives us:
\((a^2 + b^2 + c^2)^2 = a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2 + c^2a^2)\)

We now have two equations:
\( (a^2 + b^2 + c^2)^2 = a^4 + b^4 + c^4 + 2(a^2b^2 +b^2c^2+ c^2a^2)\)

\((a^2 + b^2 + c^2)^2 = 4(a^2b^2 + b^2c^2 + c^2a^2)\)

Combining the two equations gives us:
\(a^4 + b^4 + c^4 + 2(a^2b^2 + b^2c^2+ c^2a^2) = 4(a^2b^2 + b^2c^2 + c^2a^2)\)

\(a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2)\)

\(2(a^4 + b^4 + c^4) = 4(a^2b^2 + b^2c^2 + c^2a^2) = (a^2 + b^2 + c^2)^2.\)

Thus, \(2(a^4 + b^4 + c^4)\) is a perfect square.

Since condition 2) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 1) is easy to work with, and condition 2) is difficult to work with. For CMT 4(B) questions, D is most likely the answer.

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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 \(1\) variable (\(A\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)
\(A = (n-1)^2+n^2+(n+1)^2\)

\(= (n – 1)(n – 1) + n^2 + (n + 1)(n + 1)\)

\(= n^2 – n – n + 1 + n^2 + n^2 + n + n + 1\)

\(= 3n^2 + 2\)

Squares of integers do not have the remainder 2 when it is divided by 3, for the following reason:
Case 1) \(n = 3k\)

\(n^2 = (3k)^2 = 9k^2 = 3(3k^2) + 0\)

Case 2) \(n = 3k+1\)

\(n^2 = (3k+1)^2 = 9k^2 + 9k + 1 = 3(3k^2+3k) + 1\)

Case 3) \(n = 3k+2\)

\(n^2 = (3k+2)^2 = 9k^2 + 12k + 4 = 3(3k^2+4k+1) + 1\)

Thus, A is not a square, A is an irrational number, and the answer is ‘no’.

Since 'no' is also a unique answer, according to CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
\(0^2 = 0, 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2=25, 6^2 = 36, 7^2 = 49, 8^2 = 64, 9^2 = 81.\)

Thus, squares do not have a units digit 2 and the answer is ‘no’.

Since 'no' is also a unique answer, according to CMT (Common Mistake Type) 1, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT 4(B) questions, D is most likely the answer.

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.