Bunuel wrote:

SOLUTION

If the positive integer x is a multiple of 4 and the positive integer y is a multiple of 6, then xy must be a multiple of which of the following?

I. 8

II. 12

III. 18

(A) II only

(B) I and II only

(C) I and III only

(D) II and III only

(E) I, II, and III

First of all notice that we are asked: "\(xy\) MUST be a multiple of which of the following", not COULD be a multiple.

\(x\) is a multiple of 4 --> \(x=4m\), for some positive multiple \(m\), so \(x\) could be: 4, 8, 12, ...

\(y\) is a multiple of 6 --> \(y=6n\), for some positive multiple \(n\), so \(y\) could be: 6, 12, 18, ...

So, \(xy=(4m)*(6n)=24mn\), hence \(xy\) is in any case a multiple of 24, which means it must be a multiple of 8 and 12, but not necessarily of 18. For example the least value of \(x\) is 4 and the least value of \(y\) is 6, so the least value of \(xy\) is 24, which is a multiple of both 8 and 12, but not 18.

Answer: B.

Hope it's clear.

Hi....a silly question:

"if 6 is a multiple of a number then 12 is also a multiple of that number" True or false?

x: 6n--> 6,12,18,24

since 18 is not a multiple of 12, i say false. but the book answer is true.

can you please explain?

(this question is from the O-levels math book)