[GMAT math practice question]

When the positive integer n is divided by 8, the remainder is 3, and when n is divided by 5, the remainder is 2. What is the remainder when the smallest possible value of n is divided by 6?

A. 0
B. 1
C. 2
D. 3
E. 4

=>

If n has remainder 3 when it is divided by 8, then n = 8a + 3 for some integer a. The possible positive integer values of n are 3, 11, 19, 27, 35, … .
If n has remainder 2 when it is divided by 5, then n = 5b + 2 for some integer b. The possible positive integer values of n are 2, 7, 12, 17, 22, 27, 32, … .
The smallest value of n that appears in both lists is 27. Therefore, the smallest possible value for n is 27.
27 = 6*4 + 3, 27 has remainder 3 when it is divided by 6.

Therefore, the answer is D.
Answer: D
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[GMAT math practice question]

If the sequence {An} satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An}?

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

=>

We determine the period of the sequence by examining its terms:
A1 = 0 and A2 = 1.
A3 = A2 – A1 = 1 – 0 = 1
A4 = A3 – A2 = 1 – 1 = 0
A5 = A4 – A3 = 0 – 1 = -1
A6 = A5 – A4 = -1 – 0 = -1
A7 = A6 – A5 = -1 – (-1) = 0
A8 = A7 – A6 = 0 – (-1) = 1
A9 = A8 – A7 = 1 – 0 = 1
A10 = A9 – A8 = 1 – 1 = 0
…

The sequence has period 6.
The sum of the first six terms is A1 + A2 + … A6 = 0 + 1 + 1 + 0 + (-1) + (-1) = 0.
We apply this fact to determine the sum of the first 100 terms of the sequence:
A1 + A2 + … A100
= ( A1 + A2 + … A6 ) + … + ( A91 + A92 + … A96 ) + A97 + A98 + A99 + A100
= 0 + … + 0 + A97 + A98 + A99 + A100
= A97 + A98 + A99 + A100
= 0 + 1 + 1 + 0 = 2.

Therefore, the answer is B.
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[GMAT math practice question]

A certain characteristic in a large population has a distribution that is symmetric about the mean m. 68 percent of the distribution lies within one standard deviation d of the mean. The average score of students in a certain class is 72 pts. 84% of students in the class score less than 80 pts. What is the standard deviation of the students’ scores?

A. 1
B. 2
C. 4
D. 8
E. 16

=>



80 – 72 = 8 is the standard deviation of the distribution.


Therefore, D is the answer.
Answer: D
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[GMAT math practice question]

If n is an integer, which of the following must also be an integer?

I. n(n+1)/2
II. n(n+1)(n+2)/6
III. n(n+1)(n+2)(n+3)/8

A. I only
B. II only
C. III only
D. I and II
E. I, II and III

=>

Statement I
Since n and n + 1 are consecutive integers, n(n+1) is a multiple of 2.
Thus, n(n+1)/2 is an integer.

Statement II
Since n and n + 1 are consecutive integers, n(n+1) and n(n+1)(n+2) are multiples of 2.
Since n, n + 1 and n + 2 are three consecutive integers, n(n+1)(n+2) is a multiple of 3.
Thus, n(n+1)(n+2) is a multiple of 6, and n(n+1)(n+2) / 6 is an integer.

Statement III
Since n and n + 1 are two consecutive integers, n(n+1) is a multiple of 2.
Similarly, (n+2)(n+3) is a multiple of 2.
Also, either n and n + 2 or n + 1 and n + 3 are consecutive even integers. Thus, either (n + 1)(n+3) is a multiple of 8 or n(n+2) is a multiple of 8 since one of them is a multiple of 4.
It follows that n(n+1)(n+2)(n+3) is a multiple of 8, and n(n+1)(n+2)(n+3)/8 is an integer.

Therefore, the answer is E.

Answer: E
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[GMAT math practice question]

Alice travels 120 km, the first 40 km at x km/h and the remaining 80 km at y km/h. What is her average speed for the entire trip?

A. 2xy/(2x+y)
B. 2xy/(3x+y)
C. 3xy/(2x+y)
D. xy/(2x+y)
E. xy/(x+2y)

=>

The average speed is (Total Distance) / (Total Time).
The time for the first 40 km is 40 / x and the time for the remaining 80 km is 80 / y. Thus, the total time for the trip is ( 40/x + 80/y ).
Her average speed for the trip is
120 / ( 40/x + 80/y ) = 120xy / ( 40y + 80x ) = 3xy / ( y + 2x ) = 3xy /
(2x +y).

Therefore, the answer is C.
Answer : C
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[Math Revolution GMAT math practice question]

There are 3 ways to make the number 12 using products of two positive integers. These are 1*12, 2*6, and 3*4. In how many ways can 2700 be written as the product of two positive integers?

A. 6
B. 12
C. 15
D. 18
E. 36

=>

2700 = 2^2*3^3*5^2
The number of distinct factors of 2700 is (2+1)(3+1)(2+1) = 36.
Since the order of multiplication does not matter (i.e. 30 * 90 = 90*30), the number of pairs of positive integers that multiply to give 2700 is 36/2 = 18.

Therefore, the answer is D.
Answer: D
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[Math Revolution GMAT math practice question]

What is the remainder when 7^8 is divided by 100?

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

=>

The remainder when 7^8 is divided by 100 is equal to the final two digits of 7^8.
Now, 7^1 = 7, 7^2 = 49, 7^3 = 343, and 7^4 = 2401.
So, the final two digits of 7^n have period 4:
The tens digits are 0 -> 4 -> 4 -> 0
and the units digits are 7 -> 9 -> 3 -> 1.
It follows that the tens and units digits of 7^8 are 0 and 1, respectively.
Therefore, the remainder when 7^8 is divided by 100 is 1.

Therefore, the answer is A.
Answer : A
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[Math Revolution GMAT math practice question]

If m=-1 and n = 1^2 + 2^2 + … + 10^2, what is the value of m^n+m^{n+1}+m^{n+2}+m^{n+3}?

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

=>

m^n+m^{n+1}+m^{n+2}+m^{n+3}
= m^n(1+m^1+m^2+m^3)
= (-1)^n(1+(-1)^1+(-1)^2+(-1)^3)
= (-1)^n* 0 = 0
Whatever the value of n is, the answer is 0.

Therefore, C is the answer.

Answer: C
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[Math Revolution GMAT math practice question]

If a=2, b=-2, and c=3, what is the value of a^4b^{-3}c^0?

A. -2
B. -1
C. 0
D. 2
E. 3

=>

a^4b^{-3}c^0
= 2^4*(-2)^{-3}*3^0
= -(2^4*2^{-3}*1)
= -2

Therefore, the answer is A.

Answer: A
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[Math Revolution GMAT math practice question]

If the range of set A is 5 and the range of set B is 11, which of the following CANNOT be the range of sets A and B combined?

A. 10
B. 11
C. 12
D. 13
E. 14

=>

For example, if A = {0,5} and B = { 0, 11 }, then the set A is included in the set B and the smallest possible range of the sets A and B combined is 11, which is the range of the set B.
Thus, any number less than 11 cannot be the range of sets A and B, combined.

Therefore, A is the answer.
Answer: A
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