Let *r* and *s *be two rational numbers. We want to find the smallest integer *k* such that both *k/r* and *k/s* are integers. In other words we are looking for the least common multiple (denominator) of two rationals (fractions).

We can write *r *and *s *as *m/p *and *n/q* respectively, where m, p, n, and q are integers.

We have:

The second equality merits some explanation, and requires that we show two things. First that *r* and *s* both divide *lcm(p,q)*, and second that *lcm(m,n) *is the smallest such integer with this property.

For the first property we can see that,

must be an integer since *m *must divide any multiple of itself. A similar argument can be made for *s, *and so *lcm(**m,n)* divides both *r *and *s.*

For the second property let’s assume there is an integer smaller than *lcm(m,n)* that *r *and *s *both divide*. *Then there must also be an integer smaller than *lcm(m,n)* that both *m *and *n *divide, but this contradicts the definition of *lcm(m,n)*.

To summarize. The least common integer multiple of two rationals is the least common multiple of their numerators.

This all seems a bit trite, but strangely I could not find it anywhere when I looked for it, so here it is…