Out of a quite large set of random integer numbers, I selected only those that were multiples of M or N, and rejected the rest. In the resulting (obviously smaller) subset, 50% of the numbers were multiples of N. Curiously, M wasn't 50% of N.

What is the minimum possible pair of values for M and N? The next such pair? Are there infinite possible such pairs?

PS. If you don't know the reason for the "Cherry picking" title, check this reference or this other one.

PPS. And if "a quite large set" of numbers doesn't satisfy you, imagine an infinite set, with every possible integer.

Let's say GCD(M,N)=L; then, M=Lm and N=Ln, with GCD(m,n)=1. Also, LCM(M,N)=Lmn. Out of the numbers from 1 to Lmn, n are multiples of Lm (=M) and m are multiples of Ln (=N). However, the total number of multiples is m+n-1 because the number Lmn itself is counted twice. We are told that n/(m+n-1)=1/2, so n=m-1.

The minimum answer is found for L=1, n=2 (N=1 isn't allowed) and m=3.

The general answer is (Ln,L(n+1)) for integer L and n.

Posted by e.g. on 2006-06-17 22:09:05