The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

When the positive integer n is divided by 8, the remainder is 3, and when n is divided by 5, the remainder is 2. What is the remainder when the smallest possible value of n is divided by 6?

A. 0
B. 1
C. 2
D. 3
E. 4

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If n has remainder 3 when it is divided by 8, then n = 8a + 3 for some integer a. The possible positive integer values of n are 3, 11, 19, 27, 35, … .
If n has remainder 2 when it is divided by 5, then n = 5b + 2 for some integer b. The possible positive integer values of n are 2, 7, 12, 17, 22, 27, 32, … .
The smallest value of n that appears in both lists is 27. Therefore, the smallest possible value for n is 27.
27 = 6*4 + 3, 27 has remainder 3 when it is divided by 6.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

70% of students at a school joined the soccer club and 60% joined the baseball club. What percent of students joined both clubs?

1) 10% of students joined neither of the clubs.
2) 20 students joined both clubs

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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.

For questions related to 2x2 matrices and percentages, we need 3 percentages for sufficiency. Since the original condition gives us 2 percentages, we need 1 additional percentage to solve the problem.

As condition 1) gives information about a percentage, it is sufficient.

As condition 2) does not give information about a percentage, it is not sufficient.

Therefore, the answer is A.

Answer: A

[GMAT math practice question]

[x] is the least integer greater than or equal to x. If x is not an integer, what is the value of [x]?

1) The greatest integer less than x is 1
2) The nearest integer to x is 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.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

[x] can be defined in the following way:
If n – 1 < x ≤ n, where n is an integer, then [x] = n.

Condition 1)
We have 1 < x < 2.
Thus, [x] = 2.
Condition 1) is sufficient, since we have a unique solution.

Condition 2)
If x = 0.1, then [x] = 1.
If x = -0.1, then [x] = 0.
Since we do not have a unique answer, condition 2) is not sufficient.

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 m and n are positive integers, what is the value of m + n?

1) m^2< 5
2) m – n = 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.

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)
Condition 1) tells us that m = 1 or m = 2.
If m = 1, then n = 1 – 1 = 0, which is not a positive integer.
If m = 2, then n = 2 - 1 = 1.
Thus, the unique values are m = 2 and n = 1, and m + n = 3.
Both conditions 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)
Since it provides no information about the variable n, condition 1) is not sufficient.

Condition 2)
If m = 2 and n = 1, then m + n = 3.
If m = 3 and n = 2, then m + n = 5.
Since we don’t have a unique solution, condition 2) 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]

If n is positive integer, is 35 a factor of n?

1) 35 is a factor of n^2
2) 35 is a factor of 5n

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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)
Since 35=5*7 is a factor of n^2, 5 and 7 are factors of n^2.
Since 5 and 7 are prime integers and n^2 is divisible by 5 and 7, n is divisible by both 5 and 7.
Thus, 35 is a factor of n.
Condition 1) is sufficient.

Condition 2)
If n = 35, then 35 is a factor of n.
If n = 7, then 35 is not a factor of n.
Since we don’t have a unique answer, condition 2) is not sufficient.

Therefore, A is the answer.

Answer: A

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.

[Math Revolution GMAT math practice question]

Is 1/x>1/y?

1) x < y
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 may be modified as follows:
1/x > 1/y
=> xy^2 > x^2y by multiplication by x^2y^2
=> xy^2 - x^2y > 0
=> xy(y-x) > 0

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)
Applying both conditions together yields y > x > 0. So, x>0, y>0 and y-x>0.
It follows that the product xy(y – x) is positive. Therefore, both conditions are sufficient when considered together.

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 = 2 and y = 3, then 1/x = 1/2, 1/y = 1/3, and the answer is ‘yes’.
If x = -2 and y = 3, then 1/x = -1/2, 1/y = 1/3, and the answer is ‘no’.
Condition 1) is not sufficient on its own.

Condition 2)
This condition provides us with no information about the variable y, so it 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) provides 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.

[Math Revolution GMAT math practice question]

If p is a prime number and n is a positive integer, what is the number of factors of 3^np^2?

1) n = 4
2) p > 4

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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 a question asks for a number of factors, it is very important to check that all of the given prime factors are “different”. By condition 2), p is a prime number different from 3. To determine the number of factors, we need to know the exponents in the prime number factorization. Therefore, we also need condition 1).

Since p is a different prime integer from 3, and n = 4, the number of factors of 3^np^2 is (4+1)(2+1) = 15.
Since we have a unique solution, both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

[Math Revolution GMAT math practice question]

There are 3 ways to make the number 12 using products of two positive integers. These are 1*12, 2*6, and 3*4. In how many ways can 2700 be written as the product of two positive integers?

A. 6
B. 12
C. 15
D. 18
E. 36

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2700 = 2^2*3^3*5^2
The number of distinct factors of 2700 is (2+1)(3+1)(2+1) = 36.
Since the order of multiplication does not matter (i.e. 30 * 90 = 90*30), the number of pairs of positive integers that multiply to give 2700 is 36/2 = 18.

Therefore, the answer is D.
Answer: D

Why did you divide by 2 i.e. \(36/2 = 18\)?

Hi ammuseeru







I am not an expert, but here are my two cents. Hope you have not overlooked highlighted text.
The reason we divided by 2 is that the pairs can also be interchanged within themselves. It is same
as 36C2.

[Math Revolution GMAT math practice question]

If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, what is the value of y?

A. -2
B. -1
C. 0
D. 1
E. 2

=>

(xy)^2 + x^2 – 2xy – 2x + 2 = 0
=> (xy)^2 – 2xy + 1 + x^2 – 2x + 1 = 0
=> (xy – 1)^2 + (x – 1)^2 = 0
=> xy = 1 and x = 1.
Thus , x = 1 and y = 1.

Therefore, the answer is D.
Answer: D


[Math Revolution GMAT math practice question]

1. (inequality) Is x>0?
1) x = |x|
2) x2 - 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.

Condition 1)
x = |x|
=> x ≥ 0
In inequality questions, the law “Question is King” tells us that if the solution set of the question does not include the solution set of a condition, then the condition is not sufficient.
Condition 1 is not sufficient.

Condition 2)
x2-x =0
=> x(x-1) = 0
=> x = 0 or x = 1
If x = 0, then the answer is “no”.
If x = 1, then the answer is “yes”.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) & 2)
The two conditions give x = 0 or x = 1.
They are not sufficient, when taken together, by the argument above.

Therefore, the answer is E.

Answer: E

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.

[Math Revolution GMAT math practice question]

If the average (arithmetic mean) height of Bob, John and Tom is 180 cm, what is their median height?

1) Bob’s height is 175cm.
2) John’s height is 180cm.

=>

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.

When we have 3 numbers, if one of them is equal to their average, then it is also equal to their median.

Thus, condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

[Math Revolution GMAT math practice question]

When a positive integer n is divided by 3, what is the remainder?

1) When n is divided by 5, the remainder is 2
2) When n is divided by 6, the remainder is 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.

Since the divisor (6) of condition 2) is a multiple of the divisor (3) of the question, condition 2) is sufficient.

When we encounter questions related to remainders, plugging in numbers is suggested.

Condition 1)
If n = 2, then the remainder when n is divided by 3 is 2.
If n = 7, then the remainder when n is divided by 3 is 1.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
The possible values of n are
n = 2, 8, 14, 20, …
All of these values have a remainder of 2 when they are divided by 3.
Thus, condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

[Math Revolution GMAT math practice question]

If m=-1 and n = 1^2 + 2^2 + … + 10^2, what is the value of m^n+m^{n+1}+m^{n+2}+m^{n+3}?

A. -2
B. -1
C. 0
D. 1
E. 2

=>

m^n+m^{n+1}+m^{n+2}+m^{n+3}
= m^n(1+m^1+m^2+m^3)
= (-1)^n(1+(-1)^1+(-1)^2+(-1)^3)
= (-1)^n* 0 = 0
Whatever the value of n is, the answer is 0.

Therefore, C is the answer.

Answer: C

[Math Revolution GMAT math practice question]

If a=2, b=-2, and c=3, what is the value of a^4^{b-3}c^0?

A. -2
B. -1
C. 0
D. 2
E. 3

=>

a^4b^{-3}c^0
= 2^4*(-2)^{-3}*3^0
= -(2^4*2^{-3}*1)
= -2

Therefore, the answer is A.

Answer: A

[Math Revolution GMAT math practice question]

a, b, c, d, e and f are integers. Is their median greater than their average (arithmetic mean)?

1) a<b<c<d<e<f
2) b-a=d-c=f-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.

Since we have 6 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.

If a = 1, b = 2, c =5, d = 6, e = 7, f = 8, we have a median of 5.5 and an average of 29/6. The median is greater than the average, and the answer is “yes”.

If a = 1, b = 2, c =3, d = 4, e = 5, f = 6, we have a median of 3.5 and an average of 3.5. The median is not greater than the average, and the answer is “no”.

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

Therefore, the answer is E.

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.