The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

(number properties) For positive integers a, b, c, d and e, what is the tuple (a, b, c, d, e)?

1) ab=72, bc=108
2) cd=60, de=500

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 5 variables (a, b, c, d and e) 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)
Since ab = 2^3*3^2, bc = 2^2*3^3, cd = 2^2*3*5 and de = 2^2*5^3, b is a common factor of 2^3*3^2 and 2^2*3^3, c is a common factor of 2^2*3^3 and 2^2*3*5, and d is a common factor of 2^2*3*5 and 2^2*5^3.
So, b is a factor of 2^2*3^2, c is a factor of 2^2*3 and d is a factor of 2^2*5.
Two possible solutions are a=2, b=2^2*3^2, c=3, d=2^2*5 and e=5^2, and a=4, b=2*3^2, c=2*3, d=2*5 and e=2*5^2.
Since both conditions together do not yield a unique solution, 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]

(number properties) What is the value of x?

1) the prime factorization of x is ab(10a+b) (a, b are positive integers less than or equal to 9)
2) x is 4-digit 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 3 variables (x, a and b) and 1 equation, x = ab(10a+b), 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) and 2)
Since a and b are prime numbers less than or equal to 9 and 10a + b is a prime number, the possible values of a and b are a = 2 and b = 3; a = 3 and b = 7; a = 5 and b = 3; and a = 7 and b = 3.
If a = 2 and b = 3, then x = 2*3*23 = 138, which is not a 4-digit number.
If a = 3 and b = 7, then x = 3*7*37 = 777, which is not a 4-digit number.
If a = 5 and b = 3, then x = 5*3*53 = 795, which is not a 4-digit number.
If a = 7 and b = 3, then x = 7*3*73 = 1533, which is a 4-digit number.
x = 1533 is the unique solution, and 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)
Since a and b are prime numbers less than or equal to 9 and 10a + b is a prime number, the possible values of a and b are a = 2 and b = 3; a = 3 and b = 7; a = 5 and b = 3; and a = 7 and b = 3.
If a = 2 and b = 3, then x = 2*3*23 = 138.
If a = 3 and b = 7, then x = 3*7*37 = 777.
If a = 5 and b = 3, then x = 5*3*53 = 795.
If a = 7 and b = 3, then x = 7*3*73 = 1533.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Condition 2) is obviously not sufficient on its own.

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.

[GMAT math practice question]

(set) For sets A and B, define A*B to be {(a, b)|a∈A and b ∈ B}. If the number of elements of A∪B is 8 and A∩B has 4 elements, what is the maximum number of elements of A*B?

A.25
B.32
C.36
D. 42
E. 49

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Define n(X) to be the number of elements of a set X.
Since n(A∪B) = n(A) + n(B) – n(A∩B), we have n(A) + n(B) = n(A∪B) + n(A∩B) = 8 + 4 = 12.
Since n(A*B) = n(A)*n(B), the maximum value of n(A)*n(B), where n(A) + n(B) = 12, is 6*6 = 36.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(number properties) What is the value of 2a/(a+1)+2b/(b+1)?

1) a, b are integers
2) ab=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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

Since 2a/(a+1) + 2b(b+1) = (2a*(b+1) + 2b(a+1))/(a+1)(b+1) = (2ab + 2a + 2ab + 2b) / (ab+a+b+1) = (4ab + 2a + 2b) / (ab+a+b+1), the question asks the value of (4ab + 2a + 2b) / (ab+a+b+1).

Condition 1) is obviously not sufficient as it tells us nothing about the values of a and b.

Condition 2):
If ab = 1, then (4ab + 2a + 2b) / (ab+a+b+1) = (2a+2b+4) / (a+b+2) = 2(a+b+2)/(a+b+2) = 2. Condition 2) is sufficient.

Therefore, the answer is B.
Answer: B

[GMAT math practice question]

(absolute value) On the number line, the mid-point of the two integers x and y is 4. What is the value of y?

1) the absolute value of x is 6
2) x < y

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.


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

Since the mid-point of x and y is 4, (x+y)/2 = 4, and x + y = 8.

Since we have 2 variables (x and y) and 1 equation, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
Since |x|=6, we must have x=6 or x=-6.
Since 4 lies midway between x and y, we must have either x=6 and y=2, or x=-6 and y=14.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
There are many possible pairs of integers, x and y, satisfying condition 2) and the original condition.
Examples are x=1 and y=7, and x=2 and y=6.
Condition 2) is not sufficient since it does not yield a unique solution.

Conditions 1) & 2)
Condition 1) tells us that x=6 and y=2, or x=-6 and y=14.
Since condition 2) tells us that x<y, we must have x=-6 and y=14.
Both conditions together are sufficient, since they yield a unique solution.

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.

[GMAT math practice question]

(number properties) a, b and c are integers. What is the value of a-b-c?

1) |a|<|b|<|c|
2) a*b*c=12

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 3 variables (a, b and c) 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):
If a=1, b=2 and c=6, then |a| < |b| < |c|, abc = 12 and a-b-c=-7.
If a=-1, b=-2 and c=6, then |a| < |b| < |c|, abc = 12 and a-b-c=-5.

Both conditions together are not sufficient, since they don’t yield a unique solution.

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]

(algebra) Using the formula a^2–b^2 =(a+b)(a-b), find the value of (1+ 1/2 )(1+ 1/4)(1+1/16).

A. 255/128
B. 255/256
C. 257/128
D. 127/128
E. 555/128

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Let X = (1+1/2)(1+1/4)(1+1/16).
Then
(1-1/2)X = (1-1/2)(1+1/2)(1+1/4)(1+1/16)
=(1-1/4)(1+1/4)(1+1/16)
=(1-1/16)(1+1/16)
=1-1/256
=255/256.
So, (1/2)X = 255/256, and X = 255/128

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(number properties) n/2 is the cube of a positive integer and n/3 is the square of a positive integer. What is the minimum possible value of n?

A. 423
B. 432
C. 442
D. 447
E. 532

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We have n/2 = a^3 and n/3 = b^2 for some integers a and b.
So, n=2a^3 and n=3b^2 for some integers a and b.
The possible values of n=2a^3 are 2*1^3, 2*2^3, 2*3^3, 2*4^3, 2*5^3, 2*6^3, … which are 2, 16, 54, 128, 250, 432, … .
The possible values of n=3a^2 are 3*1^2, 3*2^2, 3*3^2, 3*4^2, 3*5^2, 3*6^2, 3*7^2, 3*8^2, 3*9^2, 3*10^2, 3*11^2, 3*12^2,…, which are 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 363, 432, … .
The first number to appear in both lists is 432.
Thus, the minimum possible value of n is 432.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(number properties) (k, p) denotes the remainder when k is divided by p. What is the value of (mn, 5)?

1) (m, 5)=4
2) (n, 5)=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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

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)
We have m = 5a + 4 and n = 5b + 2 for some integers a and b.
Then we have mn = (5a+4)(5b+2) = 25ab + 10a + 20b + 8 = 5(5ab+2a+4b+1)+3 and mn has a remainder 3 when it is divided by 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)
We have m = 5a + 4 from condition 1)
If we have m = 4 and n = 1, then mn = 4 has a remainder 4 when it is divided by 5.
If we have m = 4 and n = 2, then mn = 8 has a remainder 3 when it is divided by 5.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
We have m = 5b + 2 from condition 1)
If we have m = 1 and n = 2, then mn = 2 has a remainder 2 when it is divided by 5.
If we have m = 4 and n = 2, then mn = 8 has a remainder 3 when it is divided by 5.
Since condition 2) does not yield a unique solution, it 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]

(algebra) What is the value of (x+y)(2x-y)-(x-y)(2x+y)?

1) x=3/2
2) xy=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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

The expression of the question (x+y)(2x-y)-(x-y)(2x+y) is equivalent to 2xy as below.
(x+y)(2x-y)-(x-y)(2x+y)
= 2x^2 + xy – y^2 – (2x^2 –xy –y^2)
= 2xy
Thus, condition 2) is sufficient since 2xy = 2*2 = 4.

Condition 1)
Since we don’t have any information about the variable y, condition 2) is not sufficient, obviously.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(function) A function f(x) satisfies f(x)f(y)=f(x+y)+f(x-y) for every real numbers x, y. What is the value of f(0)f(1)f(2)f(3)?

1) f(0)=2
2) f(1)=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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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

If x = 1, y = 0, then we have f(1)f(0) = f(1) + f(1) = 2f(1).
If x = 1, y = 1, then we have f(1)f(1) = f(2) + f(0)
If x = 2, y = 1, then we have f(2)f(1) = f(3) + f(1)

Since we have 4 variables (f(0), f(1), f(2) and f(3)) and 3 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
If f(1) = 1, then we have f(2) = f(1)f(1) – f(0) = 1 – 2 = -1, f(3) = f(2)f(1) – f(1) = (-1)*1 – 1 = -2 and f(0)f(1)f(2)f(3) = 2*1*(-1)(-2) = 4.
If f(1) = 2, then we have f(2) = f(1)f(1) – f(0) = 4 – 2 = 2, f(3) = f(2)f(1) – f(1) = 2*1 – 2 = 0 and f(0)f(1)f(2)f(3) = 2*2*(2)*0 = 0.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since f(1) = 1, we have f(0) = f(1)f(0) = f(1) + f(1) = 2, f(2) = f(1)f(1) – f(0) = 1 – 2 = -1, f(3) = f(2)f(1) – f(1) = (-1)*1 – 1 = -2 and f(0)f(1)f(2)f(3) = 2*1*(-1)*.(-2) = 4.
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]

(number properties) What is the number of 2 digits positive integers which are relatively prime to 100?

A. 24
B. 36
C. 48
D. 51
E. 63

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Remind that the number of terms of an arithmetic sequence with its first term F, its last term L and its difference d is (L-F)/d + 1.
We have 90 of 2 digits positive integers from 10 to 99.
The number of multiples of 2 is 45 = (98-10)/2 + 1 since they are 10, 12, …, 98.
The number of multiples of 5 is 18 = (95-10)/5 + 1 since they are 10, 15, …, 95.
The number of multiples of both 2 and 5 is 9 = (90-10)/10 + 1 since they are 10, 20, …, 90.

Thus, the number of 2 digits positive intergers which are relatively prime is 90 – (45 + 18 - 9) = 36.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

(number properties) A 5-digits positive integer ab3cd is a multiple of 225. What is maximum number?

A. 98325
B. 97325
C. 96320
D. 95325
E. 95320

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Since 225 is a product of 9=3^2 and 25=5^2, its last two digits is a multiple of 25 and the sum of its all digits a + b + 3 + c + d is a multiple of 9.
98325 and 95325 are only two multiples of 25.
Since 9 + 8 + 3 + 2 + 5 = 27 is a multiple of 9, 98325 is a multiple of 9 and 97325 is not a multiple of 9 since 9 + 7 + 3 + 2 + 5 is 26 and 26 is not a multiple of 9.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

(number properties) If a, b, and c are positive integers, what is the value of a+b+c?

1) a, b and c are three consecutive odd numbers in that order
2) 20≤bc-ab≤24

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 3 variables (a, b, and c) and 0 equations, E is most likely 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)
Since a, b, and c are three consecutive odd numbers in that order, we can put a = b – 2 and c = b + 2.
bc – ab = b(b+2) – (b-2)b = b^2 + 2b – b^2 +2b = 4b and we have 20≤4b≤24 and dividing everything by 4 we get 5≤b≤6.
Since b is an odd integer, we have b=5.
Then we have a = 3, b = 5, c = 7 and a + b + c = 15.
Since both conditions together yield a unique solution, they 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)
Since there are many possibilities for (a,b,c), the condition is obviously not sufficient.

Condition 2)
If a = 3, b = 5 and c = 7, then we have a + b + c = 15.
If a = 1, b = 5 and c = 5, then we have a + b + c = 11.
Since condition 2) does not yield a unique solution, it is not sufficient.

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 when the answer is A, B, C or D.

[GMAT math practice question]

(algebra) What is the value of xyz?

1) x+ 1/y = 2
2) y – 1/z = 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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)

Since x + 1/y = 2, we have x = 2 – 1/y = (2y-1)/y.
Since y – 1/z = 1/2, we have 1/z = y – 1/2 = (2y-1)/2 or z = 2/(2y-1).
Then we have xyz = [(2y-1)/y]*y*[2/(2y-1)] = 2.
Since both conditions 1) & 2) together yield a unique solution, they are sufficient.

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]

(algebra) If [x, y, z]=(z+x)/(2z-y), then what is [3, -1, 2]?

A. -3
B. -1
C. 1
D. 3
E. 5

=>


We have [3, -1, 2] = (2+3)/(2*2-(-1)) = 5/5 = 1 by substituting in x = 3, y = --1 and z = 2.

Therefore, C is the answer.
Answer: C

[GMAT math practice question]

(probability) 50 students took a quiz with two questions A, B and each question has 10 points. 35 students solved question A correctly and 20 students solve both questions correctly. If the total average is 13 points, then how many students solved only question B correctly?

A. 4
B. 6
C. 8
D. 10
E. 12

=>

Assume x is the number of students who solved only question B correctly.
Then we have the total average (10*35 + 10x)/50 = 13 which is equivalent to
350 + 10x = 650.
=> 10x = 300
=> x = 30.

Then the number of students who solved only question B correctly is
x – 20 = 30 – 20 = 10.

Therefore, D is the answer.
Answer: D