The Ultimate Q51 Guide [Expert Level]

[GMAT math practice question]

How many numbers between 1 and 200, inclusive, have a 2 in the units place and are divisible by 4?

A. 8
B. 9
C. 10
D. 11
E. 12

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In order for a number with 2 in the units place to be divisible by 4, the last two digits must be one of 12, 32, 52, 72 and 92.
Thus, the numbers between 1 and 200, inclusive, satisfying this property are
12, 32, 52, 72, 92, 112, 132, 152 and 172.

There are 8 such numbers. Therefore, A is the answer.
Answer: A

[GMAT math practice question]

How many integers between 18 and 3399 are multiples of 17?

A. 198
B. 199
C. 200
D. 201
E. 202

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18 = 17*1 + 1, 34 = 17*2 is the least multiple of 17 greater than 18.
3399 = 17*199 + 16, 3383 = 17*199 is the greatest multiple of 17 less than 3399.
Thus, the multiples of 17 between 18 and 3399 are 34=17*2, … , 3383 = 17*199. There are 198 = (199 – 2) + 1 of these numbers as they are in one-to-one correspondence with the integers between 2 and 199, inclusive.

Therefore, the answer is A.
Answer: A

[GMAT math practice question]

What is the value of (2^{2020} + 2^{2018}) / (2^{2020}-2^{2018})?

A. 3/5
B. 5/3
C. 2^{2018}
D. 2^{2019}
E. 2^{2020}

=>

(2^{2020} + 2^{2018}) / (2^{2020}-2^{2018})
= (2^22^{2018} + 2^{2018}) / (2^22^{2018}-2^{2018})
= (2^{2018})(2^2+1) / (2^{2018})(2^2-1) = (2^2+1) / (2^2-1)
= 5/3

Therefore, the answer is B.
Answer: B

If a and b are integers, and x and y are positive integers, is ax+by >0?

1) ax+y>0
2) bx+y>0

Answer: E

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 4 variables (a, b, x and y) 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 = 1, x = 1 and y = 1, then a^x+b^y = 2, and the answer is ‘yes’.
If a = -1, b = -1, x = 1 and y = 1, then a^x+b^y = -2, and the answer is ‘no’.

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

Therefore, E is the answer.

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.

If a and b are positive integers, is a+b divisible by 7?

1) a – b is divisible by 7
2) a+b+7 is divisible by 91

Answer: B

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 2) tells us that a + b + 7 = 91k for some integer k, and so a + b = 91k – 7 = 7(13k-1). Therefore, a + b is a multiple of 7.
Thus, condition 2) is sufficient.

Condition 1)
If a = 14 and b = 7, then a + b = 21 is a multiple of 7, and the answer is ‘yes’.
If a = 8 and b = 1, then a + b = 9 is not a multiple of 7, and the answer is ‘no’.
Thus, condition 1) is not sufficient since it does not yield a unique solution.

Therefore, B is the answer.

a, b, and c are positive integers. Is a+b+c an odd number?

1) ab is an odd number
2) c is an odd number

Answer: C

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 4 variables (a, b, x and y) 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 with the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Condition 1), ab is an odd integer, is equivalent to the statement that both a and b are odd numbers.
As c is also an odd number, a + b + c is an odd number since it is the sum of three odd numbers.

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 a = 1, b = 1, and c = 1, then a + b + c = 3 is an odd number, and the answer is ‘yes’.
If a = 1, b = 1, and c = 2, then a + b + c = 4 is not an odd number, and the answer is ‘no’.
Thus, condition 1) is not sufficient.

Condition 2)
If a = 1, b = 1, and c = 1, then a + b + c = 3 is an odd number, and the answer is ‘yes’.
If a = 2, b = 1, and c = 1, then we have a + b + c = 4 is not an odd number, and the answer is ‘no’.
Thus, condition 2) is not sufficient.

Therefore, C is the answer.

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.

A box contains 1 red ball, 3 green balls, 5 yellow balls, 7 blue balls, 9 white balls, and 11 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 7 balls of the same color are drawn?

A. 7
B. 14
C. 15
D. 27
E. 28


Answer: E

The maximum number of draws without 7 balls of a single color is 1 + 3 + 5 + 6 + 6 + 6 = 27, obtained by drawing 1 red ball, 3 green balls, 5 yellow balls, 6 blue balls, 6 white balls and 6 black balls. If we draw one more ball, then we must have 7 balls of one color.
Thus, we need to draw 28 balls to ensure that 7 balls of the same color are drawn. Therefore, E is the answer.

[GMAT math practice question]

What is the largest digit n for which the number 123,45n is divisible by 3?

A. 3
B. 5
C. 6
D. 7
E. 9

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Recall that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. The sum of the digits of 123,45n is 1 + 2 + 3 + 4 + 5 + n = n + 15. This is divisible by 3 exactly when n is divisible by 3.
So, n must be a multiple of 3.
Thus, the largest digit, n is 9.

Therefore, the answer is E.
Answer: E

[GMAT math practice question]

For positive integers m and n, is m^n a perfect square?

1) The five-digit integer, 12,3m0 is a multiple of 4
2) The five-digit integer, 23,4n5 is a multiple of 9

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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 if n is an even integer or m is a perfect square. Condition 2):
“23,4n5 is a multiple of 9” is equivalent to the statement that 2 + 3 + 4 + n + 5 = n + 14 is a multiple of 9. For this to occur, we must have n = 4 and m^n = m^4 = (m^2)^2 is a perfect square. Condition 2 is sufficient.

Condition 1)
“12,3m0 is a multiple of 4” is equivalent to the statement that m is an even integers, since this is what is required for 12,3m0 to be a multiple of 4.
Thus, condition 1) tells us that m = 0, 2, 4, 6 or 8. Since we don’t know the exponent n, condition 1) is not sufficient.

Therefore, B is the answer.

Answer: B

[GMAT math practice question]

n is a positive integer. What is the remainder when n is divided by 3?

1) n^2 has remainder 1 when it is divided by 3
2) n has remainder 7 when it is divided by 9

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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)
n could be any of the integers 1, 2, 4, 5, 7, 8, …
If n is one of 1, 4, 7, then n has a remainder 1 when it is divided by 3.
If n is one of 2, 5, 8, then n has a remainder 2 when it is divided by 3.
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
n = 9k +7 can be expressed as n = 9k + 7 = 9k + 6 + 1 = 3(3k+2)+1. Therefore, n has remainder 1 when it is divided by 3.
Thus, condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

[GMAT math practice question]

Is x = y?

1) x ≤ y
2) |x| ≥ |y|

=>

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 x =1, and y = 1, then x and y satisfy both conditions, and the answer is “yes” since x = y.
If x = -2, and y = 1, then x and y satisfy both conditions, but the answer is “no” since x ≠ y.

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

Therefore, E is the answer.
Answer: E

[GMAT math practice question]

(b-a)/ab = 1/a – 1/b. What is the value of 1/2 + 1/6 + 1/12 + … + 1/90?

A. 4/5
B. 5/6
C. 8/9
D. 9/10
E. 11/12

=>

1/2 + 1/6 + 1/12 + … + 1/90
= 1/(1*2) + 1/(2*3) + 1/(3*4) + … + 1(9*10)
= (2-1)/(1*2) + (3-2)/(2*3) + (4-3)/(3*4) + … + (10-9)/(9*10)
= (1/1 – 1/2) + (1/2 – 1/3) + (1/3-1/4) + … + (1/9 – 1/10)
= 1/1 – 1/10
= 9/10

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

How many whole numbers between 100 and 400, inclusive, contain the digit 2?

A. 100
B. 125
C. 138
D. 145
E. 150

=>

There are 19 such numbers with hundreds digit 1: 102, 112, 120, 121, …, 129, 132, 142, …, 192.
There are 100 such numbers with hundreds digit 2: 200, 201, 202, 203, …, 299.
There are 19 such numbers with hundreds digit 3: 302, 312, 320, 321, …, 329, 332, 342, …, 392.

Thus, there are a total of 19 + 100 + 19 = 138 such numbers.

Therefore, the answer is C.
Answer: C


[GMAT math practice question]


When 32 is divided by k, the remainder is k-3. What is the value of k?

1) k>20
2) k<40

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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 original condition says that 32 = k*q + ( k – 3 ) or 35 = k*q + k = k(q+1).
That is, k is a factor of 35.
So, k = 1, 5, 7 or 35.

Since condition 1) “k>20” yields the unique solution “k=35”, condition 1) is sufficient.

Condition 2) yields k = 1, 5, 7 or 35. Since it does not yield a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

[GMAT math practice question]

Is the average (arithmetic mean) of x, y, and z equal to their median?

1) y is the average (arithmetic mean) of x, y, and z
2) z = 2y - x

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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 data are symmetric, then the average and the median of the data are the same.

Condition 1):
If y is the mean of x, y and z, then x, y, z are symmetric. To see this note that ( x + y + z ) / 3 = y implies that y = ( x + z ) / 2. So, 2y = x + z and y – x = z – y, which means that x, y, z are symmetric.
Condition 1) is sufficient.

Condition 2):
z = 2y – x is equivalent to y – x = z – y, which is equivalent to condition 1).
Condition 2) is sufficient.

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.

Therefore, D is the answer.
Answer: D

[GMAT math practice question]

If n is a positive integer, is n a prime number?

1) n is the least factor of 77 greater than 1
2) n has exactly two factors

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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)
The factors of 77 are 1, 7, 11 and 77. We have n = 7 since the least factor of 77 other than 1 is 7.
Condition 1) is sufficient.

Condition 2)
n is a prime number, by definition, since it has exactly two factors.
Condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

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]

How many committees can be formed comprising 2 male members selected from 4 men, 3 female members selected from 5 women, and 3 junior members selected from 6 juniors?

A. 900
B. 1200
C. 1500
D. 1800
E. 2400

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There are 4C2 ways to select 2 men from 4 men, 5C3 ways to select 3 women from 5 women and 6C3 ways to select 3 juniors from 6 juniors. Therefore, the total number of possible committees is
4C2*5C3*6C3 = {(4*3)/(2*1)}{(5*4*3)/(3*2*1)}{(6*5*4)/(3*2*1)} = 6*10*20 = 1200.

Therefore, B is the answer.
Answer: B