Overview of GMAT Math Question Types and Patterns on the GMAT

[GMAT math practice question]

(number properties) If x, y, and z are integers, is xyz a multiple of 6?

1) x+y+z is a multiple of 6
2) x, y, and z are consecutive

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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 the product of three consecutive integers is a multiple of 6, condition 2) is sufficient. Without loss of generality, since condition 2) tells us that x, y and z are consecutive, we can write y = x + 1 and z = x + 2. Then xyz = x(x+1)(x+2) is a multiple of 6 since x(x+1) is a multiple of 2 and x(x+1)(x+2) is a multiple of 3.

Condition 1)
If x = 1, y = 2 and z = 3, then xyz = 6 and the answer is ‘yes’.
If x = 2, y = 2 and z = 2, then xyz = 8 and the answer is ‘no’.
Condition 1) is not sufficient, since it doesn’t yield a unique answer.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(geometry) Is triangle ABC a right triangle?

1) The perimeter of triangle ABC is 12
2) The length of the shortest side of triangle ABC is 3

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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 questions about triangles require 3 variables, 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.

Let x, y and z be the side-lengths of triangle ABC, and suppose x ≤ y ≤ z.
Then x = 3 and x + y + z = 12.
If y = 4 and z = 5, triangle ABC has side lengths 3, 4 and 5. Thus it is a right triangle, and the answer is ‘yes’.
If y = 4.5 and z = 4.5, triangle ABC has side lengths 3, 4.5 and 4.5. It is not a right triangle, and the answer is ‘no’.
Since the two conditions don’t yield a unique answer when applied together, they 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]

The sum of two numbers is 1 and their product is -1. What is the sum of their cubes?

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

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Let the numbers be x and y. Then x + y = 1 and xy = -1.
Since (x+y)^2 = x^2 + 2xy + y^2 = x^2 + y^2 – 2, we have x^2 + y^2 = 3,
and x^3 + y^3 = (x+y)(x^2-xy+y^2) = 1*(x^2+ 1 + y^2) = x^2+y^2+1 = 4.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

(number properties) If k/mn, where k, m and n are positive integers, is a fraction in its lowest terms, is k/mn a terminating decimal?

1) 1/m is a terminating decimal
2) 1/n 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.

In order for k/mn to a terminating decimal, mn must have no prime factors other than 2 and 5. This implies that neither m nor n have prime factors other than 2 and 5. Thus, we need both conditions 1) & 2) together.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(number properties) n is a positive odd integer. What is the value of n?

1) n, n + 2, n + 4 are prime numbers.
2) n - 1 is an even prime 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.

Condition 1):
The triple 3, 5 and 7 is the unique triple made up of three consecutive odd prime numbers number. So, n = 3.
Thus, condition 1) is sufficient.

Condition 2):
Since n – 1 = 2 is the unique even prime number, we must have n = 3. Thus, condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

[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]

(number properties) If m and n are positive integers, is m/n a terminating decimal?

1) m is divisible by 9
2) n is divisible by 30

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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 = 30, then m/n = 9 / 30 = 3/10 = 0.3 is a terminating decimal and the answer is ‘yes’.
If m = 9 and n = 210, then m/n = 9 / 210 = 3/70 is not a terminating decimal since the denominator, 70, has a prime factor other than 2 and 5. The answer is ‘no’.
Both conditions together are not sufficient, since they don’t yield a unique solution.

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]

(inequality) x is a positive number. Is x > 1?

1) √x > x
2) x^3 - 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.

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.

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

Condition 1)
√x > x
=> x > x^2 by squaring
=> x^2 – x < 0
=> x(x-1) < 0
=> 0 < x < 1
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 1) is sufficient.

Condition 2)
x^3 - x < 0
=> x(x^2-1) < 0
=> (x+1)x(x-1) < 0
=> x<-1 or 0 < x < 1
=> 0 < x < 1 since x is positive.
Since ‘no’ is also a unique answer by CMT (Common Mistake Type) 1, condition 2) is sufficient.


Therefore, D is the answer.
Answer: D

[GMAT math practice question]

(number property) If a=2b=6c for positive integers a, b, and c, which of the following could be the value of abc?

A. 1000
B. 1230
C. 2250
D. 2367
E. 2488

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Since 2b = 6c, we have b = 3c. Thus, abc = (6c)*(3c)*c = 18c^3 and abc is a multiple of 18. This implies that abc is a multiple of both 2 and 9.

Of the above answer choices, only 2250 is a multiple of both 2 and 9.

Therefore, the answer is C.
Answer: C

[GMAT math practice question]

(number properties) x = -1. What is the value of x + x^n + x^{n+1} + x^{n+2}?

1) n is a prime number
2) n is an odd 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 condition on its own first.

Condition 1)
If n = 2, then x + x^n + x^{n+1} + x^{n+2} = (-1) + (-1)^2 + (-1)^3 + (-1)^4? = (-1) + 1 + (-1) + 1 = 0.
If n = 3, then x + x^n + x^{n+1} + x^{n+2} = (-1) + (-1)^3 + (-1)^4 + (-1)^5 = (-1) + (-1) + (-1) + 1 = -2.
Condition 1) is not sufficient since it doesn’t yield a unique solution.

Condition 2)
If n is an odd number, then x + x^n + x^{n+1} + x^{n+2} = (-1) + (-1)^n + (-1)^{n+1} + (-1)^{n+2} = (-1) + (-1) + 1 + (-1) = -2.
Condition 2) is sufficient since it yields a unique solution.

Therefore, B is the answer.
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]

(inequality) When -1/2<h<0, which of the following has the greatest value?

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

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Test each expression by plugging in h = -1/4.
h^2-2h+1 = (h-1)^2 = (-(5/4))^2 = 25/16
h^2+h = h(h+1) = -(1/4)(3/4) = -(3/16)
h = -1/4
h^2 = (-(1/4))^2 = 1/16
h^3 = (-(1/4))^3 = -(1/64)

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(exponent) m and n are positive numbers. (2^m)^n = 512. What is the minimum value of m + n?

A. 3
B. 4
C. 5
D. 6
E. 8

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(2^m)^n = 2^{mn} = 512 = 2^9.
Thus mn = 9.
Recall that if mn is a fixed constant, and m and n are positive numbers, then m + n has its minimum value when m = n.
Thus, we must have m = n = 3 to obtain the minimum value of m + n = 3 + 3 = 6.

Therefore, the answer is D.
Answer: D

[GMAT math practice question]

(number properties) Is pqr is a multiple of 5?

1) p, q and r are consecutive odd integers.
2) p, q and r 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 3 variables (p, q and r) 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):
3, 5 and 7 is the unique triplet of three consecutive odd integers which are prime numbers, and 3*5*7 is a multiple of 5.
Thus, 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)
If p = 3, q = 5 and r = 7, then pqr = 105 is a multiple of 5, and the answer is ‘yes’.
If p = 7, q = 9 and r = 11, then pqr = 693 is not a multiple of 5, and the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
If p = 3, q = 5 and r = 7, then pqr = 105 is a multiple of 5, and the answer is ‘yes’.
If p = 3, q = 7 and r = 11, then pqr = 231 is not a multiple of 5, and the answer is ‘no’.
Condition 2) is not sufficient since it doesn’t yield a unique solution.

Therefore, C is the answer.
Answer: C

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]

(absolute value) Is x<y<z ?

1) |x+2|<y<z+2
2) |x-2|<y<z-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.

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)
Now, by condition 1), x < x + 2 ≤ | x + 2 | < y < z + 2, and x < y.
By condition 2), |x-2|<y<z-2 < z, and so y < z.
Thus, both conditions together are sufficient since they yield x < y < z, and the unique answer is ‘yes’.

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 = 4, and z = 7, then the answer is ‘yes’.
If x = 1, y = 4, and z = 3, then the answer is ‘no’.
Condition 1) is not sufficient since it doesn’t yield a unique answer.

Condition 2)
If x = 1, y = 4, and z = 7, then the answer is ‘yes’.
If x = 5, y = 4, and z = 7, then the answer is ‘no’.
Condition 2) is not sufficient since it doesn’t 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]

(function) If the symbol # represents one of addition, subtraction, multiplication, or division, what is the value of (2 # 1)?

1) 2 # 2 = 1
2) (–1/2) # (-1) = 1/2

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

Condition 1)
# is the division operation since 2 / 2 = 1.
So, 2#1 = 2/1 = 2.
Condition 1) is sufficient since it yields a unique answer.

Condition 2)
# could be the multiplication, subtraction or the division operation
since (-1/2) * (-1) = 1/2, (-1/2)-(-1) = (1/2) and (-1/2)/(-1) = 1/2.

Since 2 * 1 = 2 and 2 / 1 = 2, but 2-1 = 1, condition 2) does not yield a unique value for 2#1.
Condition 2) is not sufficient since it doesn’t yield a unique answer.

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.

Please help me to solve this
Two people run the circle with 10 rounds, one runs in the inside and the other runs in the outside. When the difference between the inside and the outside is 15 feet, what is the difference of between two people's distance after running 10 rounds, approximately?
A. 800 B. 850 C. 900 D. 950 E. 1,000

The sequence An satisfies An = A_(n-1)/A_(n-2) , where n is a positive integer greater than 2. If A1=1 and A2=2, then A123=?

A. 1/2
B. 1/4
C. 1
D. 2
E. 4

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A1 = 1
A2 = 2
A3 = A2/A1 = 2/1 =2
A4 = A3/A2 = 2/2 =1
A5 = A4/A3 = 1/2
A6 = A5/A4 = (1/2)/1 = 1/2
A7 = A6/A5 = (1/2)/(1/2) =1
A8 = A7/A8 = 1/(1/2) = 2
Since A7 = 1 and A8 = 2, the sequence is periodic with period 6.
Thus, A123 = A3 = 2 since 123=6•20+3

Therefore, the answer is D.
Answer: D