Look at the following infinite sequence: 1, 10, 100, 1000, 10000, . . .. What is the proof that there are 2 numbers in this sequence that differ by a multiple of 12345678987654321?
Let $n = 12345678987654321$.  Consider the first $n+1$ elements of the powers-of-10 sequenc modulo $n$.  According to the pigeonhole principle,  there are at least two of these elements that have the same residue modulo $n$.  Their difference is a multiple of $n$.
Take the digit 1 from the 10 and take the digit 1 from the 100.  They differ by 0 which is a multiple of 12345678987654321.