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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