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 and as m/p and n/q respectively, where m, p, n, and q are integers.

We have:

\displaystyle \text{lcm}(r,s) = \text{lcm} \Bigg(\frac{m}{p},\frac{n}{q}\Bigg) = \text{lcm}(m,n)

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,

\displaystyle \frac{\text{lcm}(m,n)}{\frac{m}{p}} = \text{lcm}(m,n)\frac{p}{m}

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

For the second property let’s assume there is an integer smaller than lcm(m,n) that  and both divideThen there must also be an integer smaller than lcm(m,n) that both and 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…

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