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.
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…