The multiples of 12 and of 21 are written in order. What is the smallest positive difference between a multiple of 12 and a multiple of 21?

Well    12 x 21 = 252      so the smallest possible difference is ZERO.

If you want the smallest possible POSITIVE difference, let's write out the multiples

12 24 36 48 60 72 84 96 108 120 132 144 156 168

    21      42    63    84 105    126          147       168    This repeats every 4th multiple of 21 (or 7th multiple of 12)

The LEAST POSITIVE difference is 3 it appears .

The multiples of 12 and of 21 are written in order.

What is the smallest positive difference between a multiple of 12 and a multiple of 21?

The difference is \(n\cdot 21 -m \cdot 12 = d \) with \( n,m,d \in Z\) has a solution,

if the \(gcd(21,12)\) is a divider of \(d\). Or \(gcd(21,12) | d\).

The greatest common divisor  \(gcd(21,12) = 3.\)

The difference is \(n\cdot gcd(21,12)\)

so the difference is a multiple of 3

The differences are: \(0, 3, 6, 9, 12, \ldots\)

The smallest positive difference is 3.