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21 Mar 2022, 06:05
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
41% (01:29) correct
59%
(01:22)
wrong
based on 78
sessions
History
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Not Attempted Yet
If x, y, and z are integers, is x/(yz) an integer?
(1) y is a factor of x more than once.
(2) All of the prime factors of z are also factors of y.
(1) y is a factor of x more than once.
(2) All of the prime factors of z are also factors of y.
25 Mar 2022, 14:56
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Bunuel
Breaking Down the Info:
Note, even if x is a multiple of y and x is a multiple of z, \(\frac{x}{yz}\) still might not be an integer. (E.g., x = 12, y = 4, z = 6).
Statement 1 Alone:
This means we have \(y^2\) as a factor of x. We don't know z, so insufficient.
Statement 2 Alone:
z could have the same prime factors of y but have a higher power of each prime. So we don't know who is a factor of who, and they might not even be multiples of each other. (E.g. \(z = 2^3*3^2, y = 2^2*3^3\)Insufficient.
Both Statements Combined:
We could have \(z = y^3\) and x might not be divisible by z. Insufficient.
Answer: E
vverma1995
GMAT 1: 750 Q50 V41
Posts: 144
29 Mar 2022, 01:57
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Bunuel
Here's how I approached it:
Solution required:
x/(yz) is an integer if both y and z are factors of x together, and not just individually
Information Given:
Option 1: y is a factor of x more than once
This implies x = c * y^a where c is some other constant product of the remaining factors of x and a is an integer > 1
BUT
we know nothing about z, so the option is NOT SUFFICIENT
Option 2: All of the prime factors of z are also factors of y.
_____
An important concept to know here:
Consider the number 36. On prime factorization we get:
36 = 2 * 2 * 3 * 3
When we say "ALL PRIME FACTORS" we mean not just the two unique prime factors (2 and 3) but also the repetitions of these unique prime factors.
So, all prime factors = 2,2,3,3
____
With this in mind, z is a factor of y
However, we don't know the relationship of x with z or y, hence option 2 is NOT SUFFICIENT
Option 1 and 2 together
From 2, we know that z is a factor of y
From 1, we know that x = c * y^a where a is an integer >1
On combining 1 and 2, we know that yz is a factor of x, therefore x/(yz) is an integer.
Both together are SUFFICIENT.
Answer is C
