If m and n are positive integers, and if p and q are differe

If m and n are positive integers, and if p and q are different prime numbers, do p/q and n/m represent different numbers?

Given: \(n\) and \(m\) are integers \(p\) and \(q\) are different prime numbers.

Question: is \(\frac{n}{m}=\frac{p}{q}\)?

Notice that, since \(n\) and \(m\) are integers then this equation will hold true if \(n\) and \(m\) are multiples of prime numbers \(p\) and \(q\) respectively. For example: \(\frac{n}{m}=\frac{2}{3}=\frac{4}{6}=\frac{6}{9}=\frac{p}{q}=\frac{2}{3}\). If we were not told that \(p\) and \(q\) are different then this won't be necessary, for example following case would be possible: \(\frac{n}{m}=\frac{8}{8}=1=\frac{3}{3}=\frac{p}{q}\)

(1) Neither n nor m is a prime number. If \(n=px\) and \(m=qx\) (for some integer x more than 1) then the answer ill be YES, if not then the answer will be No. For example: if \(p=2\), \(q=3\) and \(n=2*4=8\), \(m=3*4=12\) then \(\frac{n}{m}=\frac{p}{q}=\frac{2}{3}\). Not sufficient.

(2) m is not dvisible by q. As discussed above, if \(m\) is not a multiple of \(q\) then \(\frac{n}{m}\neq{\frac{p}{q}}\). Sufficient.

Answer: B.