The Ultimate Q51 Guide [Expert Level]

If x, y, and z are positive integers, is x+y divisible by 7?

1) x+z is divisible by 7
2) y+z is divisible by 7

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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.

Conditions 1) & 2):
If x = 7, y = 7, and z = 7, the answer is ‘yes’, since x + y = 14 is divisible by 7.
If x = 1, y = 1, and z = 6, the answer is ‘no’, since x + y = 2 is not divisible by 7.
Thus, conditions 1) and 2) together are not sufficient.

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. 

[GMAT math practice question]

Which of the following is closest to 11^5*9^5–7*10^7?

A. 10^2
B. 10^7
C. 10^8
D. 10^9
E. 10^10


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Since 11 is close to 10 and 9 is close to 10, 11^5*9^5 is close to 10^5*10^5 = 10^10.
Since 7*10^7 is a relatively small number compared with 10^10, 11^5*9^5 – 7*10^7 is approximately equal to 10^10.

Therefore, the answer is E.

Answer: E

[GMAT math practice question]

For positive integer m and n, is 2m/3n an integer?

1) m is a multiple of 3
2) n is a multiple of 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.

Since we have 2 variables (m and n) 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):
m = 3, n = 2: 2m/3n = 6/6 = 1 is an integer.
m = 3, n= 4: 2m/3n = 6/12 = 1/2 is not an integer.

Since we don’t have a unique solution, both conditions together are not sufficient.

Therefore, E is the 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.

[GMAT math practice question]

What is the number of factors of a positive integer n?

1) n is a multiple of 7
2) n is a prime number

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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 of the conditions on their own first.

Condition 1)
n = 7 : The number of factors of n is 2.
n = 14 : The number of factors of n is 4.
Since the answer is not unique, condition 1) is not sufficient.

Condition 2)
n = p^1, where p is a prime number.
The number of factors on n is 1 + 1 = 2.
Thus, condition 2) is sufficient.

Therefore, the answer is B.

Answer: B

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]

What is the value of x+y?

1) x+y is between 50 and 60
2) x and y are prime numbers

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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 = 23, y = 29 : x + y = 52
x = 23, y = 31 : x + y = 54

Since we don’t have a unique solution, both conditions together are not sufficient.

Therefore, the answer is E.

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.

[GMAT math practice question]

For the integers x and y, if xy is a multiple of 25, is y a multiple of 5?

1) x-y is a multiple of 5
2) x is a multiple of 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, and then recheck the question.

Since xy = 25 =5^2, x is a multiple of 5 or y is a multiple of 5. If x is a multiple of 5, y = x – 5 is also a multiple of 5. Thus the condition 1) is sufficient.

Condition 2)
Since we don’t have any information from the condition 2), it is not sufficient.

Therefore, A is the answer.

Answer: A

[GMAT math practice question]

Which of the following is 6/7 times as far from 7/3 as is 17/6 from 2/3?

A. 88/21
B. 21/88
C. 77/20
D. 20/77
E. 78/21

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We have |x – 7/3| = (6/7)|17/6 – 2/3|.
|x - 7/3| = (6/7)(13/6)
x – 7/3 = ±(6/7)(13/6) = ±13/7
x = ±13/7 + 7/3
x = (39 + 49)/21 = 88/21 or x = (-39 + 49)/21=10/21

Therefore, the answer is A.

Answer: A

[GMAT math practice question]

Is x^y <0?

1) x is positive
2) y is a prime number

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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.

Question:
x^y <0
⇔ x < 0 and y is an odd integer.

Condition 1)
Since x > 0, x^y > 0, regardless of the value of y.
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
If x = -1 and y = 3, x^y = (-1)^3 = -1 < 0, and the answer is ‘yes’.
If x = -1 and y = 2, x^y = (-1)^2 = 1 > 0, and the answer is ‘no’.
Condition 2) is not sufficient since the question does not have a unique answer.

Therefore, A is the answer.

Answer: A

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]

If n!/(n-2)!<100, what is the greatest possible value of n?

A. 8
B. 9
C. 10
D. 11
E. 12

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We must have
n! / (n-2)! = n(n-1) < 100.
If n = 10, n(n-1) = 10*9 = 90 < 100.
If n = 11, n(n-1) = 11*10 = 110 > 100.

10 is the greatest value of n for which n!/(n-2)! < 100.

Therefore, C is the answer.

Answer: C

[GMAT math practice question]

If x and y are integers, is x a multiple of 5?

1) x+y is a multiple of 5
2) x+2y is a multiple of 5

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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 with the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2):
x = 2(x+y) – (x+2y) is a multiple of 5 since 2(x+y) and (x+2y) are multiples of 5.
Thus, both conditions 1) & 2) together are sufficient.

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)
If x = 5 and y = 5, x is a multiple of 5, and the answer is ‘yes’.
If x = 2 and y = 3, x is not a multiple of 5, and the answer is ‘no’.
Thus condition 1) is not sufficient.

Condition 2)
If x = 5 and y = 5, x is a multiple of 5, and the answer is ‘yes’.
If x = 1 and y = 2, x is not a multiple of 5, and the answer is ‘no’.
Thus condition 2) is not sufficient.

Therefore, the answer is C.

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]

If x=y-z, is x>y?

1) y<0
2) z<0

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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 can modify the question as follows:
x>y
⇔ x-y > 0
⇔ -z > 0 by the original condition
⇔ z < 0
This is the same as condition 2).

Condition 1)
If x = -1, y = -2, and z = -1, x > y, and the answer is ‘yes’.
If x = -2, y = -1, and z = 1, x < y, and the answer is ‘no’.
Condition 1) is not sufficient since the question does not have a unique answer.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

If neither x nor y is divisible by 3, which of the following could be the value of x^2+y^2?

A. 333
B. 334
C. 335
D. 336
E. 337

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Consider the squares of the integers that are not divisible by 3:
1^2 = 1, 2^2 = 4, 4^2 = 16, 5^2 = 25, 7^2 = 49, 8^2=64, ….
They all have a remainder of 1 when they are divided by 3.
Thus, the sum of the squares of two integers which are not divisible by 3 must have a remainder of 2 when it is divided by 3.

The only answer choice having this property is 335.

Therefore, C is the answer.

Answer: C

[GMAT math practice question]

If x + y > 0, is xy^2 + x^2y > 0?

1) x > y
2) xy > 1

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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.

Question:
xy^2 + x^2y > 0
=> xy(x+y) > 0
=> xy > 0, since x + y > 0
So, the question asks if xy > 0.

Since xy > 1 > 0 can be derived from condition 2), it is sufficient.
Condition 1) gives us no information about the sign of xy.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

Is (x-1)^3< (x-1)?

1) x>-1
2) x<0

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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.

The question can be modified as follows:
(x-1)^3< (x-1)
=> (x-1)^3- (x-1) < 0
=> (x-1)((x-1)^2- 1) < 0
=> (x-1)((x-1)+1)((x-1)-1) < 0
=> x(x-1)(x-2) < 0
=> x < 0 or 1 < x < 2
So, the question asks whether x < 0 or 1 < x < 2.

Condition 1)
Since the set of the question doesn’t include that of condition 1), it is not sufficient.

Condition 2)
Since the set of the question includes that of condition 2), it is sufficient.

Therefore, the answer is B.

Answer: B

[GMAT math practice question]

What is the median value of the data displayed in the following frequency table?



1) x=2
2) y=5

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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 y = 1, the median is x.
If y = 2, the median is ( x + 3 ) / 2
If 3 ≤ y ≤ 9, the median is 3.
If y = 10, the median is ( 3 + 4 ) / 2 = 3.5
If y ≥ 10, the median is 4.

Thus, the condition 2) is sufficient.

Therefore, the answer is B.

Answer: B

[GMAT math practice question]

If r, s are positive integers, is r/s a terminating decimal?

1) 1/r^2 is a terminating decimal
2) 1/s^2 is a terminating decimal

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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 can modify the question as follows:
r/s is a terminating decimal if and only if s has no other prime factors than 2 and 5.
This is the same as condition 2). Condition 1) tells us nothing about the prime factors of s.

Therefore, B is the answer.


Answer: B


[GMAT math practice question]

If (n+2)!= n!(an^2+bn+c), then abc=?

A. 2
B. 3
C. 4
D. 6
E. 8

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(n+2)! = (n+2)(n+1)n! = (n^2 + 3n + 2)n!
So,
a = 1, b = 3, and c = 2.
Thus, abc = 6.

Therefore, D is the answer.

Answer: D