I have been working on some problems and one of them has been particularly challenging. The problem is as follows.
Find a non-trivial (meaning more than 1 digit) positive integer a that satisfies:
$a/10 + 3121(a \mod 10) = 0 \mod a$
Here the division operator is meant to denote integer division so algebraically this is equivalent to:
$(\lfloor a/10\rfloor + 3121(a - 10*\lfloor a/10\rfloor)) - a*\lfloor\lfloor a/10\rfloor + 3121(a - 10*\lfloor a/10\rfloor))/a)\rfloor = 0$
I believe there should be some straightforward way to calculate this value. Using a brute force guess and check I found that $a = 101$ satisfied this congruence but I wanted to know if there was a way to analytically work this one out?