Number Theory: Least Value of Product

$500 = 2^2 \times 5^3$. For $N^N$ to be divisible by $2^2 \cdot 5^3$, $N = a+b$ must be a multiple of $2 \times 5 = 10$. Least positive integer $a+b = 10$. To minimize $a \times b$ for positive integers $a, b$ summing to 10, choose $a = 1, b = 9$. Least product $a \times b = 1 \times 9 = 9$.