The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(number properties) If n is a positive integer, is n^2+2n+44 divisible by 4?

1) n is an even integer.
2) n^2 is divisible by 144


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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.

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

Asking if “n^2+2n+44 is divisible by 4” is equivalent to asking if n is an even integer, since 44 is a multiple of 4 and n^2+2n = n(n+2) is a multiple of 4 when n is an even number.
Thus, condition 1) is sufficient.

Condition 2)
Since n^2 is divisible by 144, n is divisible by 12 and n is an even number.
Thus, condition 2) is also sufficient.

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).

[GMAT math practice question]

(number properties) What is the value of n?

1) n is the product of 2 different prime numbers less than 15
2) n and 210 are relatively prime

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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.

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Both n = 6 and n = 10 are products of two different prime numbers less than 15. Thus, condition 1) is not sufficient since it does not yield a unique solution.

Condition 2)
Both n = 11 and n = 13 are relatively prime to 210. Thus, condition 2) is not sufficient since it doesn’t yield a unique solution.

Conditions 1) & 2)
The prime numbers less than 15 are 2, 3, 5, 7, 11 and 13.
Since n and 210 = 2*3*5*7 are relatively prime, we must have n = 11*13 = 143.
Both conditions 1) & 2) together are sufficient since they yield a unique solution.

Therefore, C is the 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.

[GMAT math practice question]

(absolute value) What is the value of x?

1) |x+y|=|x-y|
2) |y| > y

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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.


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)

|x+y|=|x-y|
=> |x+y|^2=|x-y|^2
=> (x+y)^2=(x-y)^2
=> x^2+2xy+y^2=x^2-2xy+y^2
=> 4xy = 0
=> xy = 0
=> x = 0 or y = 0
Condition 1) is equivalent to x = 0 or y = 0. Condition 1) is not sufficient.
Condition 2) is equivalent to y < 0. It follows from condition 2) that x = 0 since y can’t be zero.
Both conditions 1) & 2) together are sufficient since they yield a unique solution.

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.

However, as we have seen above, neither condition 1) nor condition 2) is sufficient on its own.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(inequality) If 0＜x+1＜1, which of the following lists is written in ascending order?

A. x, x^{-1}, x^2, 1, x^{-2}
B. x^{-2}, x, x^2, 1, x^{-1}
C. x^{-1}, x, x^2, 1, x^{-2}
D. x^{-1}, x^2, x, 1, x^{-2}
E. x^{-1}, x, x^2, x^{-2}, 1

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Plug in x = -1/2.
Then x^{-2} = 4, x^{-1}= -2 and x^2 = 1/4.
Thus, x^{-1}< x < x^2 < 1 < x^{-2}.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(number properties) n is an integer. What is the units digit of n^5-5n^3+4n?

A. 0
B. 2
C. 4
D. 6
E. 8

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n^5-5n^3+4n = n(n^4-5n^2+4) = n(n^2-1)(n^2-4) = n(n+1)(n-1)(n-2)(n-2) = (n-2)(n-1)n(n+1)(n+2).
Thus, n^5-5n^3+4n is a product of five consecutive integers.
Since it is divisible by 2, 3, 4, and 5, it is a multiple of 10.
Thus, its units digits is 0.

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

(absolute value) Is x > 0?

1) |x| + |y| > |x + y|
2) |y| > y


=>

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.

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):

Condition 1) is equivalent to the requirement that xy < 0 and condition 2) is equivalent to the requirement that y < 0. Thus, both conditions together are sufficient.

Since this question is an inequality 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 = -1, then x > 0 and the answer is ‘yes’.
If x = -1, y = 1, then x < 0 and the answer is ‘no’.
Condition 1) is not sufficient, since it doesn’t yield a unique solution.

Condition 2)
Since condition 2) doesn’t provide any information about x, it is not sufficient.

Therefore, C is the 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.

[GMAT math practice question]

(number properties) If m and n are non-negative integers, what is the value of mn?

1) 3^m=5^n
2) |m|=- √n

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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.

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

Condition 1) implies that m = n = 0, since 3^m and 5^n are powers of prime numbers that are equal. Thus, condition 1) is sufficient.
Condition 2) also implies that m = n = 0, since |m|=- √n is only possible when m = n = 0 (- √n <=0 and |m| >=0). Thus, condition 2) is also sufficient.

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).

MathRevolution

What is the value of x?



Can you elaborate above? We have two variables (x,y) and 1 equation (St 1) and 1 inequality (St 2) as per my observation.

[GMAT math practice question]

(statistics) If the average (arithmetic mean) of 5 numbers is 21, what is their standard deviation?

1) The least of the 5 numbers is 21
2) The greatest of the 5 numbers is 21

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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.

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

If the maximum or minimum value of a data set is the same as its average, then all data points are equal and their standard deviation is 0.
Thus, each of the conditions is sufficient.

Therefore, D is the answer.
Answer: D

Recall the property that the SD is zero if the maximum value or the minimum value is the same as the average, as this implies that all data points are equal.
Questions related to the above property have appeared frequently on recent GMAT exams.

[GMAT math practice question]

(function) If f(x)=x^4+4x^2+7, f(x)=g(x)^2+3 and g(x) > 0, then g(x)=?

A. x^2+1
B. x^2+2
C. x^2+3
D. x^2-1
E. x^2-2

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g(x)^2 = f(x) – 3 = x^4+4x^2+7 – 3 = x^4+4x^2+4 = (x^2+2)^2
Thus, since g(x) > 0, g(x)=x^2+2.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(geometry) A rectangular board measuring 9 inches by 11 inches is to be covered with 2 inch by 3 inch rectangular tiles and 1-inch square tiles. What is the smallest number of tiles that can be used to cover the board?

A. 19
B. 20
C. 22
D. 24
E. 25

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The smallest number of tiles will be obtained by using as many 2 inch by 3 inch rectangular tiles as possible since they are the larger tiles. As shown in the diagram above, 16 such tiles can be placed on the board, leaving three square inches to be covered with 1-inch square tiles. The total number of tiles used is 16 + 3 = 19.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(absolute value) What is the value of xy?

1) x^2-4xy+5y^2-4xy+4 = 0
2) |x-4| = - √(y-2)

=>

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 conditions if necessary.

Condition 1)
x^2-4xy+5y^2-4xy+4 = 0
=> x^2-4xy+4y^2+y^2-4xy+4 = 0
=> (x-2y)^2 + (y-2)^2 = 0
=> x = 2y and y =2, since (x-2y)^2 ≥ 0 and (y-2)^2 ≥ 0.
So, x = 4 and y =2. Thus, xy = 8.
Condition 1) is sufficient since it yields a unique solution.

Condition 2)
|x-4| = - √(y-2)
=> |x-4| +√(y-2) = 0
=> |x-4| = 0 and √(y-2) = 0, since |x-4| ≥ 0 and √(y-2) ≥ 0.
So, x = 4 and y =2. Thus, xy = 8.
Condition 2) is sufficient since it yields a unique solution.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(ratio) a, b and c are numbers such that abc≠ 0 and a/b=b/c=c/d. What is the value of a/d?

1) a = 1
2) b = 2

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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.

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

a/b=b/c => b^2 = ac ⇔ a = b^2/c
b/c=c/d => c^2 = bd ⇔ d = c^2/b
So, a/d = (b^2/c) / (c^2/b) = b^3/c^3 = (b/c)^3 = (a/b)^3
Thus, conditions 1) & 2) are sufficient, when applied together.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(inequality) x, y and y are positive numbers satisfying x < y < z. Is x < 5?

1) x^2 + y^2 + z^2 = 125
2) y^2 = 25

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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.

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.

Condition 1)
If x ≥ 5, then x^2 + y^2 + z^2 > x^2 + x^2 + x^2 > 3x^2 > 3*5^2 = 125.
Thus, we must have x < 5. Condition 1) is sufficient since it yields the unique answer, ‘yes’.

Condition 2)
Since x^2 < y^2 = 25, and x and y are positive numbers, x < y = 5.
Thus, x < 5. Condition 2) is sufficient since it yields the unique answer, ‘yes’.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number property) x + 2y + 3z = 4 and 5x + 4y + 3z = 8. What is the value of x + y + z?

A. 1
B. 2
C. 3
D. 4
E. 5

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When we add the two equations, we obtain (x+2y+3z) + (5x+4y+3z) = 6(x+y+z) = 4 + 8 = 12. Dividing both sides of this equation by 6 yields x+y+z = 2.

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

(number properties) m and n are integers. Which of the following cannot be the value of (m^2+1)(n^2+1)?

A. 10
B. 20
C. 50
D. 60
E. 100


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D is likely to be the answer as it is the only one which is divisible by 3. We test (m^2+1)(n^2+1) for divisibility by 3. Note that every integer m can be written as 3k, 3k+1 or 3k+2 for some integer k.
If m = 3k, then m^2 = 9k^2, and m^2 + 1 = 9k^2 + 1 = 3(3k^2) + 1.
If m = 3k+1, then m^2 = (3k+1)^2 = 9k^2 + 6k + 1 = 3(3k^2+2k) + 1, and m^2 + 1 = 3(3k^2+2k) + 2.
If m = 3k+2, then m^2 = (3k+2)^2 = 9k^2 + 12k + 4 = 3(3k^2+4k+1) + 1, and m^2 + 1 = 3(3k^2+4k+1) + 2.
In all cases, m^2+1 is not divisible by 3. Similarly, n^2+1 is not divisible by 3. Since 3 is a prime number, this implies that (m^2+1)(n^2+1) is not divisible by 3.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number properties) If two integers have no common factors other than 1, they are called relatively prime. Are x and z relatively
prime?

1) x and y are relatively prime.
2) y and z are relatively prime.

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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.

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)
If x = 2, y = 3, z = 5, then x and z are relatively prime, and the answer is ‘yes’.
If x = 2, y = 3, z = 2, then x and z are not relatively prime, and the answer is ‘no’.

Both conditions together are not sufficient, since they don’t yield a unique answer.

Therefore, E is the 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.