GMAT® Official Guide 2019 Quantitative Review
PS01466- For positive integers a and b, the remainder when a
is divided by b is equal to the remainder when b is
divided by a. Which of the following could be a value of
ab?
I. 24
II. 30
Ill. 36(A) II only
(B) Ill only
(C) I and II only
(D) II and Ill only
(El I, II, and IllArithmetic
We are given that the remainder when a is
divided by b is equal to the remainder when b
is divided by a, and asked about possible values
of ab. We thus need to find what our given
condition implies about a and b.We consider two cases: a= band a-:/:-b.If a = b, then our given condition is trivially
satisfied: the remainder when a is divided by a
is equal to the remainder when a is divided by b.
The condition thus allows that a be equal to b.Now consider the case of a-:/:-b. Either a< b or
b < a. Supposing that a < b, the remainder when
a is divided by bis simply a. (For example, if 7 is
divided by 10, then the remainder is 7.) However,
according to our given condition, this remainder,
a, is also the remainder when b is divided by a,
which is impossible. If b is divided by a, then the
remainder must be less than a. (For example, for
any number that is divided by 10, the remainder
cannot be 10 or greater.) Similar reasoning
applies if we suppose that b < a. This is also
impossible.We thus see that a must be equal to b, and
consider the statements I, II, and III.I. Factored in terms of prime numbers,
24 = 3 x 2 x 2 x 2. Because "3" occurs only
once in the factorization, we see that there is
no integer a such that ax a = 24. Based on
the reasoning above, we see that 24 cannot
be a value of ab.II. Factored in terms of prime numbers,
30 = 5 x 3 x 2. Because there is no integer a
such that a x a = 30, we see that 30 cannot
be a value of ab.
III. Because 36 = 6 x 6, we see that 36 is a
possible value of ab (with a = b).
The correct answer is B.
PS01867- List S consists of the positive integers that are
multiples of 9 and are less than 100. What is the
median of the integers in S?
34.(A) 36
(B) 45
(C) 49
(Dl 54
(E) 63Arithmetiic
In the set of positive integers less than 100, the
greatest multiple of 9 is 99 (9 X 11) and the least
multiple of 9 is 9 (9 x 1). The sequence of positive
multiples of 9 that are less than 100 is therefore
the sequence of numbers 9 X k, where k ranges
from 1 through 11. The median of the numbers
k from 1 through 11 is 6. Therefore the median
of the numbers 9 x k, where k ranges from 1
through 11, is 9 x 6 = 54.The correct answer is D.D_ _..., _____ _._~xo
A C
F
PS07397
In the figure above, if Fis a point on the line that
bisects angle ACO and the measure of angle DCF is x^0 ,
which of the following is true of x?(A) 90 •.::; X < 100
(B) 100.::; X < 110
(C) 110 .::; X < 120
(D) 120 • .::;x<l30
(E) 130 .::; X < 140