2023 AMC 10B Problems/Problem 15

Problem

What is the least positive integer such that is a perfect square?

Solution

Consider 2, there are odd number of 2's in (We're not counting 3 2's in 8, 2 3's in 9, etc). There are even number of 3's in ...

So, original expression reduce to

\begin{align*}
m \cdot 2 \cdot 4 \cdot 6 \cdot 8 \cdot 10 \cdot 12 \cdot 14 \cdot 16 &\equiv m \cdot 2^8 \cdot (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8)\\
&\equiv m \cdot 2 \cdot 3 \cdot 5  \cdot (2 \cdot 3)  \cdot 7  \cdot (2  \cdot 2 \cdot 2)\\
&\equiv m  \cdot 2 \cdot 5  \cdot 7
\end{align*}
m &= 2 \cdot 5 \cdot 7 = 70 (Error compiling LaTeX. Unknown error_msg)