Bunuel wrote:
How many prime factors of x are also prime factors of y?
(1) x = 30
(2) y is a multiple of x
(1) x = 30
We do not know y
\(y = 30 \implies \text{shared prime factors} = \{2,5,3\}\\
y = 10 \implies \text{shared prime factors} = \{2,5\}\)
Insufficient(2) y is a multiple of x
Neither x or y are fixed. Prove by example:
\(x = y = 2 \implies \text{shared prime factors} = \{2\}\\
x = y = 6 \implies \text{shared prime factors} = \{2,3\}\)
Insufficient(1 AND 2).We have \(x\) fixed, and know that \(y\) is a multiple of \(x\).
For some multiple \(k \in \mathbb{Z}\)
\(y = kx\).
Note that (2) does not state that the multiple must be positive.
\(k = 0 \implies \text{shared prime factors} = \varnothing\\
k \neq 0 \implies \text{shared prime factors} = \text{prime factors of 30} = \{2,3,5\}\)
Insufficient(E) statements 1 and 2 together are not sufficient, and additional data is needed to answer the question
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