The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(number properties) What is the greatest common divisor of positive integers m and n?

1) m and n are consecutive
2) m^2 – 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)
The gcd of two consecutive integers is always 1.
Thus, condition 1) is sufficient.

Condition 2)
If m^2–n^2 = m+n, then (m+n)(m-n)=m+n and m-n = 1 since m+n ≠0.
This implies that m and n are consecutive integers, and their gcd is 1.
Condition 2) is sufficient since it yields a unique answer.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

[GMAT math practice question]

(number property) If p and q are prime numbers, what is the number of factors of 6pq?

1) p and q are different
2) p<3<q

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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.

Recall that if n = p^aq^br^c, where p, q and r are different prime numbers, and a, b and c are non-negative integers, then n has (a+1)(b+1)(c+1) factors.

Condition 2)
We must have p = 2 since p is prime and p < 3.
The prime factorization of 6pq is 2*3*p*q = 2^2*3*q since q is prime and q > 3.
The number of factors of 2^2*3*q is (2+1)(1+1)(1+1) = 12. Condition 2) is sufficient since it yields a unique answer.

Condition 1)
If p = 2 and q = 3, then 6pq = 2^2*3^2 and the number of factors is (2+1)(2+1)= 9.
If p = 5 and q = 7, then 6pq = 2*3*5*7 and the number of factors is (1+1)(1+1)(1+1)(1+1) = 8.
Condition 1) is not sufficient since it does not yield a unique solution.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(probability) How many rectangles are found in the lattice below?



A. 90
B. 100
C. 120
D. 150
E. 180

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Each rectangle is uniquely determined by the intersections between two vertical lines and two horizontal lines.
Since we have 6 vertical lines and 5 horizontal lines, the number of rectangles is 6C2*5C2 = 15*10 = 150.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number properties) m and n are positive integers greater than 6. What is the value of m + n?

1) m*n = 504
2) m and n are multiples of 6

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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) allows us to write m = 6a and n = 6b, where a and b are integers greater than 1.
So, m*n = 6a*6b = 36*a*b = 504.
This yields ab = 14. So, a = 2 and b = 7 or a = 7 and b = 2.
Thus, m=12 and n=42 or m=42 and n=12, and we obtain the unique answer m+n = 54.
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)
m*n = 504 = 2^3*3^2*7
If m = 12 and n = 42, then m + n = 54.
If m = 8 and n = 63, then m + n = 71.
Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If m = 12 and n = 12, then m + n = 24.
If m = 12 and n = 24, then m + n = 36.
Condition 2) is not sufficient since it does not yield a unique answer.

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) m * n = 2145, where m and n are positive integers. What is the value of m + n?

1) m and n are two-digit integers.
2) m and n both have remainder 3 when they are divided by 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.

Since we have 2 variables (m and n) and 1 equation (mn=2145), D is most likely to be the answer.

Condition 1)
m * n = 2145 = 3*5*11*13.
Since m and n are two-digit numbers, there are four cases to consider:
i) m=33 (= 3*11) and n=65 (= 5*13)
ii) m=39 (=3*13) and n=55 (=5*11)
iii) m=65 (=5*13) and n=33 (= 3*11)
iv) m=55 (=5*11) and n=39 (=3*13)
So, there are two possible values of m + n, which are 98 and 94.
Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If m = 39 and n = 55, then m + n = 94.
If m = 3 and n = 715(=5*11*13), then m + n = 718.
Condition 2) is not sufficient since it does not yield a unique answer.

Conditions 1) & 2)
m * n = 2145 = 3*5*11*13.
Condition 1) gives rise to the following four cases for the values of m and n:
i) m=33 (= 3*11) and n=65 (= 5*13)
ii) m=39 (=3*13) and n=55 (=5*11)
iii) m=65 (=5*13) and n=33 (= 3*11)
iv) m=55 (=5*11) and n=39 (=3*13)
Of these, only
m=39 and n=55, and m=55 and n=39 satisfy condition 2)
So, we have a unique answer m+n=94.
Thus, both conditions together are 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]

(function) The parabola y=f(x)=a(x-h)^2+k lies in the x-y plane. What is the value of k?

1) y=f(x) passes through (1,0) and (3,0).
2) y=f(x) passes through (2,1) and no y-value is greater than 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.



The question asks for the minimum or maximum value for the function.
Condition 2 is sufficient since it provides the maximum value for the function.

Condition 1)



The parabolas drawn above both pass through the points (1,0) and (3,0). It is obvious that we don’t have a unique maximum or minimum function value. Condition 1) is not sufficient.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(number properties) What is the largest multiple of 24 that can be written using each of the digits 0, 1, 2, … , 9 exactly once?

A. 9876543210
B. 9876543120
C. 987654102
D. 1234567890
E. 1234567980

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A number is a multiple of 24 precisely when it is a multiple of both 3 and 8.
The sum of 0, 1, 2, … , 9, “0 + 1 + 2 + … + 9 = 45” is a multiple of 3. Thus, a number which uses each of the digits 0, 1, 2, … , 9 exactly once is a multiple of 3.
In order for a number to be a multiple of 8, the last three digits of the number must be a multiple of 8. And the smallest multiple of 8 that can be formed using three of the digits 0, 1, 2, … , 9 is 120.
Thus, the biggest possible multiple of 24 is 9876543120.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(absolute value) y≠0. What is the value of x/y?

1) x^2-6xy+9y^2=0
2) |x-3|+|y-1|=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.

Note: Even though C is most likely to be an answer since we have two variables, D is likely to be the answer by Tip 1) because conditions 1) and 2) provide the same information.

Condition 1)
(x^2-6xy+9y^2)=0
=> (x-3y)^2 = 0
=> x -3y = 0
=> x = 3y
=> x/y = 3
Thus, condition 1) is sufficient.

Condition 2)
|x-3|+|y-1|=0
=> |x-3|+|y-1|=0
=> x=3 and y=1
So, x/y = 3/1 =3.
Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number properties) m and n are positive integers greater than 1. Is m^n a perfect square?

1) m is an odd integer
2) n is an odd integer

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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)
If m = 9 and n = 3, then mn = 9^3 = (3^2)^3 = 3^6 = (3^3)^2 = 27^2, which is a perfect square, and the answer is ‘yes’.
If m = 3 and n = 3, then mn = 3^3 = (3)^3 = 27, which is not a perfect square, 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.

[GMAT math practice question]

(inequality) If x, y and z are integers with x<y<z, is z>4?

1) x+y+z=12
2) x<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.

From condition 1), since x + y + z = 12, their average is 4.
The maximum of three numbers is greater than or equal to their average. Thus, z ≥4.
Indeed, we must have z > 4 since x, y and z are different for the following reasons.
If z = 4, then x + y = 8. But x < y < z (= 4), so this is impossible.
If z ≤ 4, then x < y < z ≤ 4, and we must have x < 4 and y < 4.
This implies that x + y + z < 4 + 4 + 4 = 12 and x + y + z ≠ 12 , which contradicts condition 1).
Thus, z > 4.
Condition 1) is sufficient.

Condition 2)
If x = 3, y = 4 and z = 5, then z > 4 and the answer is ‘yes’.
If x = 1, y = 2 and z = 3, then z < 4 and the answer is ‘no’.
Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

If m and n are positive integers, what is the remainder when 12^{mn} is divided by 13?

1) m=even
2) n=odd

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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.
In remainder questions, you get the same answer if you do the divisions first and calculate with the remainders or do the calculations first and then find the remainder.
Since 12 = 13*1 + (-1), 12, the remainder when 12^{mn} is divided by 13 is the same as the remainder when (-1)^{mn} is divided by 13.
If mn is an odd number, (-1)^{mn} = -1 and if mn is an even number, (-1)^{mn} = 1.
The question asks if mn is an odd number or an even number.

Condition 1)
If m is an even integer, mn is an even number and the remainder when 12^{mn} is divided by 13 is 1.
Condition 1) is sufficient, since it yields a unique answer.

Condition 2)
If m = 2 and n = 1, then 12^{2*1} = 12^2 = 144 = 13*11 + 1 and the remainder is 1.
If m = 1 and n = 1, then 12^{1*1} = 12^1 = 12 = 13*0 + 12 and the remainder is 12.
Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(geometry) What is the volume of a sphere?

1) The circumference of the sphere is 6π
2) The surface area of the sphere is 36π

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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.

Since we have 1 variable, the radius r for the sphere, and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since the circumference of the sphere is 2πr = 6π, we must have r = 3.
The volume of the sphere is (4/3)πr^2 = (4/3)π(3)^2 = 12π.
Condition 1) is sufficient since it yields a unique answer,

Condition 2)
Since the surface area of the sphere is 4πr^2 = 36π, we must have r = 3.
The volume of the sphere is (4/3)πr^2 = (4/3)π3^2 = 12π.
Condition 2) is sufficient since it yields a unique answer,

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).

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]

Is the integer n even?

1) There is a sum of n consecutive integers that is even.
2) [n/2] is an even number, where [n] is the greatest integer less than or equal to 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.

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 the sum of 2, 3, 4, 5 and 6 is 20, which is an even number, and n=5 is an odd number, the answer is sometimes ‘no’.
Since the sum of 1, 2, 3, and 4 is 10, which is an even number, and n=4 is an even number, the answer is sometimes ‘yes’.
Condition 1) is not sufficient since it does not yield a unique answer.

Condition 2)
If n = 4, then [n/2] = 2 is even, and n is even.
If n = 5, then [n/2] = 2 is also even, but n is not even.
Condition 2) is not sufficient since it does not yield a unique answer.

Conditions 1) & 2)
Since the sum of 2, 3, 4, 5 and 6 is 20, which is an even number, and n=5 is an odd number such that [n/2] = 2 is even, the answer is sometimes ‘no’.
Since the sum of 1, 2, 3, and 4 is 10, which is an even number, and n=4 is an even number such that [n/2] = 2 is even, the answer is sometimes ‘yes’.
Conditions 1) & 2) together are not sufficient, since they do not yield a unique answer.

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.

[GMAT math practice question]

Alice, Bob, Cindy, Daren, Eddie, Farrell sit on chairs around a round table. Alice sits opposite Farrell. How many seating arrangements are possible?

A. 2
B. 6
C. 24
D. 120
E. 720

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First, place Alice and Farrell in opposite chairs as shown above (there is only one way to do this as the table is round).
Next, arrange the four remaining people in 4! = 24 ways.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(algebra) 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + … = π^2/6. What is the value of 1/1^2 + 1/3^2 + 1/5^2 + … ?

A. π^2/8
B. π^2/2
C. π^2
D. 2π^2
E. 4π^2

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Set x = 1/1^2 + 1/3^2 + 1/5^2 + … . Then
π^2/6 =1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + …
= (1/1^2 + 1/3^2 + 1/5^2 + … ) + (1/2^2 + 1/4^2 + 1/6^2 + … )
= (1/1^2 + 1/3^2 + 1/5^2 + … ) + (1/(1*2)^2 + 1/(2*2)^2 + 1/(2*3)^2 + … )
= (1/1^2 + 1/3^2 + 1/5^2 + … ) + (1/4)(1/1)^2 + 1/2^2 + 1/3^2 + … )
=x+(1/4)(π^2/6)
So, x= π^2/6-(1/4)(π^2/6)=(3/4)(π^2/6)=π^2/8

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

(function) Is x + y < -1?

1) |x|>x and |y|>y
2) x^2+y^2>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 (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) tells us that x and y are negative numbers.
By condition 2), (x+y)^2 = x^2 + 2xy +^y2 > x^2 + y^2 > 1 and (x+y)^2 > 1 since x and y are negative and xy > 0.

Since x + y < 0, we must have x + y < -1.
Conditions 1) & 2) are sufficient, when considered together.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(absolute value) Is |x-1|<|x-3|?

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.

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

Now,
|x-1|<|x-3|
=> |x-1|^2<|x-3|^2
=> (x-1)^2<(x-3)^2
=> x^2-2x+1<x^2-6x+9
=> 4x < 8
=> x < 2

Thus, condition 1) is sufficient.

Condition 2)
Since the solution set of the question does not contain the solution set of condition 2), condition 2) is not sufficient.

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Therefore, A is the answer.
Answer: A