Problem of the Week
Problem C and Solution
The Smallest Part

Problem

Suppose $N=2^2\times 3^2\times 5^2\times k$, where $k$ is a positive integer. If $N$ is divisible by $2025$, then what is the smallest possible value for $k$?

Solution

First we note that $2025=3^4\times 5^2$. Then, since $N$ is divisible by $2025$, $N$ must have at least four factors of $3$ and at least two factors of $5$.

$N$ already has two factors of $3$ and two factors of $5$. Thus, $N$ needs at least two more factors of $3$ in order to make it divisible by $2025$. Therefore, the smallest possible value for $k$ is $3^2=9$.