Question

Prove that $12456$ is divisible by $36$ without actually dividing it

Answer 1

First notice that $36=4\times 9$. Hence it is enough to prove that the given number is separately divisible by $4$ and $9$ (then it will be divisible by their LCM which is $36$). Consider the number formed by the last two digits: $56$. It is divisible by $4$. Hence $12456$ is divisible by $4$. On the other hand the sum of the digits is $18$ and divisible by $9$. Hence $12456$ is divisible by $9$ as well. Combining, we get the result, $12456$ is divisible by $36$