Find all the five digit numbers in the form $65x1y$ multiples of $12$
2026-04-14 11:20:27.1776165627
Numbers $65x1y$ multiples of 12
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The number to be found can be written as $N=65010+100x+y$, where $0\le x<10$ and $0\le y<10$ are the unknown digits. Therefore $$ N=650\cdot 100+100x+10+y=4\cdot 25(650+x)+10+y $$ which is divisible by $4$ if and only if $10+y$ is divisible by $4$, so if and only if $y=\dots$ (hint: there are two cases).
The number $N$ is divisible by $3$ if and only if $$ 6+5+x+1+y $$ is divisible by $3$. Given the two choices for $y$, you'll find the possible values of $x$.