Exercise Answers
Discrete Quantitative Questions
- C Determine the prime factorization of each of the numbers.
16 = 2 × 2 × 2 × 4
24 = 2 × 2 × 2 × 3
72 = 2 × 2 × 2 × 3 × 3
Each term contains three 2s. The greatest common factor is thus 2 × 2 × 2 = 8.- 120 The LCM of a set must contain the prime factorization of each term in
the set. First, find the prime factorization of each term in the set:
6 = 2 × 3
8 = 2 × 2 × 2
15 = 5 × 3
The LCM must thus have the factors of 6: 2 × 3. The LCM must have the
factors of 8: 2 × 2 × 2. Since 6 contains a 2, the LCM will contain all the factors
of 8 and 6 when it has 2 × 2 × 2 × 3 in its prime factorization. For the LCM to
contain the factors of 15, an additional 5 is required. Thus the LCM is: 2 × 2 ×
2 × 3 × 5 = 120.- A and D Since x is divisible by 12, it must contain the prime factors of 12. The
prime factorization of 12 is 2 × 2 × 3. Since y is divisible by 8, it must contain
the prime factors of 8. The prime factorization of 8 is 2 × 2 × 2. xy must
contain all the prime factors of x and all the prime factors of y. Thus the prime
factorization of xy must contain 2 × 2 × 2 × 2 × 2 × 3. Note that xy can contain
other prime factors as well, but the ones just given are the only prime factors
that it has to contain.
Choice A: Since the prime factors of xy can be combined to yield 48, 48
must be a factor of xy, which means xy is a multiple of 48. → Choice A
is true.
Choice B: The prime factorization of 64 is 2 × 2 × 2 × 2 × 2 × 2. The
factorization 2 × 2 × 2 × 2 × 2 × 3 does not contain the prime factorization
of 64. Thus 64 is not necessarily a factor of xy. → Eliminate Choice B.
Choice C: The information in the question tells you which prime factors xy
must have, but it does not eliminate the possibility that xy has other factors.
Thus it cannot be determined whether 5 is a factor of xy. → Eliminate
Choice C.
Choice D: The prime factorization of 32 is 2 × 2 × 2 × 2 × 2. The
factorization 2 × 2 × 2 × 2 × 2 × 3 contains the prime factorization of 32.
Thus 32 is a factor of xy. → Choice D is true.
The correct answer is A and D. - A and B Since this is a “must be true” question, identify only what is
necessarily true about a. Since a is a multiple of 6 and 8, a must contain the
CHAPTER 9 ■ NUMBER PROPERTIES 18303-GRE-Test-2018_173-312.indd 183 12/05/17 11:51 am