The Ultimate Q51 Guide [Expert Level]

[Math Revolution GMAT math practice question]

If k is a positive integer and n=(k-1)k(k+1), is n a multiple of 8?

1) k is an odd number
2) k = 1

=>

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 (n and k) and 1 equation, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since k is an odd number, k – 1 and k + 1 are consecutive even integers.
Any product of consecutive even integers is a multiple of 8.
Thus, condition 1) is sufficient.

Condition 2)
Since k = 1, we have n = (k-1)k(k+1) = 0*1*2 = 0. 0 is a multiple of any number, so n = 0 is a multiple of 8.
Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

Since this question is a CMT4(B) question. Condition 2) is easy to understand and condition 1) is hard. When one condition is easy to understand, and the other is hard, D is most likely to be 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.

[Math Revolution GMAT math practice question]

If x and y are positive integers, is √15xy an integer?

1) xy is a multiple of 15
2) x and y are prime numbers

=>

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)
When we consider both conditions together, there are two sets of possible values of x and y: x = 3, y = 5 and x = 5, y = 3. In both cases, xy = 15, so
√15xy = √15*3*5 = √225 = 15 is an integer.
Thus, both conditions together are sufficient.

Since this question is a statistics 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 = 3 and y = 5, then √15xy = √15*3*5 = √225 = 15 is an integer.
If x = 6 and y = 5, then √15xy = √15*6*5 = √450 = 15√2 is not an integer.
Since we don’t have a unique answer, condition 1) is not sufficient by CMT (Common Mistake Type) 2.

Condition 2)
If x = 3 and y = 5, then √15xy = √15*3*5 = √225 = 15 is an integer.
If x = 2 and y = 5, then √15xy = √15*2*5 = √150 = 5√6 is not an integer.
Since we don’t have a unique answer, condition 2) is not sufficient by CMT (Common Mistake Type) 2.

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.

[Math Revolution GMAT math practice question]

Let A=2^{50}, B=3^{30}, and C=4^{20}. Which of the following is true?

A. A<B<C
B. A<C<B
C. B<A<C
D. B<C<A
E. C<B<A

=>

A = 2^{50} = (2^5)^{10} = (32)^{10}
B = 3^{30} = (3^3)^{10} = (27)^{10}
C = 4^{20} = (4^2)^{10} = (16)^{10}

Thus, (16)^{10} < (27)^{10} < (32)^{10} and C < B < A.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

A club has 10 members. One president and two vice-presidents are elected. In how many ways can they be selected?

A. 240
B. 280
C. 300
D. 320
E. 360

=>

The number of ways to choose one president out of 10 people is 10C1 = 10.
The number of ways to choose two vice-presidents out of the remaining 9 people is 9C2 = (9*8)/(1*2) = 36.
Thus, the number of ways to choose one president and two vice-presidents out of 10 people is 10 * 36 = 360.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

Is x + 1/x > 2?

1) x > 0
2) x ≠ 1

=>

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.

x + 1/x > 2
=> x^3 + x > 2x^2 after multiplying both sides by x^2
=> x^3 - 2x^2 + x > 0
=> x^3 - 2x^2 + x > 0
=> x(x^2 - 2x + 1) > 0
=> x(x-1)^2 > 0
=> x > 0 and x ≠ 1

Thus, we need both conditions together for sufficiency.

Therefore, C is the answer.
Answer: C

[Math Revolution GMAT math practice question]

n is a positive integer. Is n divisible by 3?

1) 36/n is divisible by 3
2) 27/n is divisible by 3

=>

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 in the original condition, D is most likely to be the answer. So, we should consider each condition on its own first. It is suggested we plug in numbers when solving remainder problems.


Condition 1)
If n = 3, then 36/3 = 12 is divisible by 3, and n is divisible by 3. The answer is ‘yes’.
If n = 1, then 36/1 = 36 is divisible by 3, but n is not divisible by 3. The answer is ‘no’.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
If n = 3, then 27/3 = 9 and n is divisible by 3. The answer is ‘yes’.
If n = 1, then 27/1 = 27 and n is not divisible by 3. The answer is ‘no’.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Even if we consider both conditions together, we still have two possible values of n: n = 1 and 3.
Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, E is the answer.
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 a and b are integers, is a-b an even number?

1) a^2b^2 is an even number
2) a^2+2b^2 is an even number

=>

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.

Modifying the question:
For a – b to be an even number, either both a and b must be even numbers or both a and b must be odd numbers.

Since we have 2 variables (a and b) 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)
From condition 2), a is an even number.
From condition 1), b might either be even or odd.

Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, the correct 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.

[Math Revolution GMAT math practice question]

If n is a positive integer, which of the following could be the value of (n+1)^3 - n^3?

A. 629
B. 630
C. 631
D. 632
E. 633

=>

Recall that x^3 – y^3 = (x-y)(x^2+xy+y^2).’
Now,
(n+1)^3 - n^3
= ( n + 1 – n ) ( (n+1)^2 + (n+1)n + n^2 )
= (n+1)^2 + (n+1)n + n^2
= n^2 + 2n + 1 + n^2 + n + n^2
= 3n^2 + 3n + 1
= 3n(n+1) + 1
Thus (n+1)^3 - n^3 has remainder 1 when it is divided by 3.
631 is the only answer choice with remainder 1 when it is divided by 3.

Therefore, the answer is C.
Answer: C

[Math Revolution GMAT math practice question]

[x] is the greatest integer less than or equal to x. What is the value of x?

1) [x] = 2
2) x is an integer

=>

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.

[x] is analyzed as follows.
If n ≤ x < n + 1 for some integer n, then [x] = n.

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

Condition 1)
[x] = 2
=> 2 ≤ x < 3
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
Since there are a lot of integers, condition 2) does not yield a unique solution. This condition is not sufficient.

Conditions 1) & 2)
x = 2 is the unique integer such that 2 ≤ x < 3.
Thus, both conditions together are sufficient.

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.

[Math Revolution GMAT math practice question]

If m and n are positive integers, is mn an even number?

1) m/n is an even number.
2) m + n is an even number.

=>

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.

Modifying the question:
mn is an even number precisely when at least one of m and n is even. So,
the question asks if either m or n is even.

Condition 1:
If m/n = 2k for some integer k, then m = 2kn, which is an even number.
Thus, condition 1) is sufficient.

Condition 2)
If m = 2 and n = 4, then m + n = 6 is even, and mn = 8 is an even number, so the answer is ‘yes’.
If m = 1 and n = 3, then m + n = 4 is even, and mn = 3 is not an even number, so the answer is ‘no’.
Since it does not give us a unique answer, condition 2) is not sufficient.

Therefore, the correct answer is A.
Answer: A

[Math Revolution GMAT math practice question]

When n is a positive integer, is n / 4 an integer?

1) n - 1 is not divisible by 2
2) n + 1 is not divisible by 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.

Modifying the question:
Asking whether n/4 is an integer is equivalent to asking whether m is a multiple of 4.

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)
Since n – 1 is not divisible by 2, n – 1 is an odd number and n is an even number.
If n = 4, then n is a multiple of 4 and the answer is ‘yes’.
If n = 2, then n is not a multiple of 4 and the answer is ‘no’.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
Since n + 1 is not divisible by 2, n + 1 is an odd number and n is an even number.
If n = 4, then n is a multiple of 4 and the answer is ‘yes’.
If n = 2, then n is not a multiple of 4 and the answer is ‘no’.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2):
If n = 4, then neither n – 1 nor n + 1 is divisible by 2, but n is a multiple of 4 and the answer is ‘yes’.
If n = 2, then neither n – 1 nor n + 1 is divisible by 2, and n is not a multiple of 4 and the answer is ‘no’.
Thus, both conditions together are not sufficient, since they do not yield a unique solution.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

What is the solution set of (1+|x|)(1+x) > 0?

A. x > -1
B. x < -1
C. x < 0
D. x > 0
E. x > 1

=>

Since 1+|x| > 0, we can divide both sides of the inequality by 1 + |x| to obtain 1+x > 0 or x > -1.

Therefore, the answer is A.
Answer: A

[Math Revolution GMAT math practice question]

How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime number?

A. 60
B. 120
C. 150
D. 180
E. 240

=>

It is easiest to use complementary counting. That is, count the number of subsets that contain no prime number and subtract it from the total number of subsets.

The number of subsets containing no prime number is the number of subsets of { 1, 4, 6, 8 }. Note that 1 is neither a prime number nor a composite number.

The number of subsets of {1,2,3,4,5,6,7,8} is 2^8 = 256.
The number of subsets of {1,4,6,8} is 2^4 = 16.
Thus, the number of subsets of {1,2,3,4,5,6,7,8} containing at least one prime number is 256 – 16 = 240.

Therefore, the answer is E.
Answer: E

[Math Revolution GMAT math practice question]

[x] is the greatest integer less than or equal to x. <x> is the least integer greater than or equal to x. What is the value of x?

1) [x] = 2
2) <x> = 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.

[x] is analyzed as follows.
If n ≤ x < n + 1 for some integer n, then [x] = n.
<x> is analyzed as follows.
If n – 1 < x ≤ n for some integer n, then <x> = n.

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

Condition 1)
[x] = 2
=> 2 ≤ x < 3
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
<x> = 2
=> 1 < x ≤ 2
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Only x = 2 satisfies both conditions.
Since the answer is unique, both conditions together are sufficient.

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.

[Math Revolution GMAT math practice question]

Is x/y>1?

1) x>y
2) x-y>1

=>

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.

x/y>1
=> xy>y^2
=> xy-y^2>0
=> y(x-y)>0
By condition 2), x-y > 1 > 0, but we can’t determine whether y is positive from condition 1).

Therefore, E is the answer.
Answer: E

[Math Revolution GMAT math practice question]

Is |m-n|=|m|-|n| ?

1) m-n = 0
2) n = 0

=>

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.

Squaring both sides of |m – n| = |m| - |n| yields
|m-n|^2=(|m|-|n|)^2
=> (m-n)^2=(|m|-|n|)^2
=> m^2+n^2-2mn=|m|^2+|n|^2-2|mn|
=> m^2+n^2-2mn=m^2+n^2-2|mn|
=> -2mn=-2|mn|
=> mn=|mn|
=> mn≥0

Condition 1):
m – n = 0 implies that m = n. So mn = n^2 ≥ 0 for all values of n.
Condition 1) is sufficient.

Condition 2):
If n = 0, then 0 = mn ≥ 0.
Condition 2) is sufficient.

Therefore, the correct answer is D.
Answer: D

[Math Revolution GMAT math practice question]

In the coordinate plane, the y-intercept of the line k is equal to twice its slope. The y-intercept is not 0. What is the x-intercept of the line k?

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

=>

The equation of line k has the form, y = mx + b, where m is the slope and b is the y-intercept.

Let m be the slope of line k. Then b = 2m, and the equation is y = mx + 2m = m(x+2).
The x-intercept of line k is -2, since -2 is a root of m(x+2)=0.

Therefore, the answer is A.
Answer: A