Dutch Mathematical Olympiad

For a three-digit positive integer, we can multiply the three digits together. We call the result the digit product of that number. For example, $123$ has digit product $1 \times 2 \times 3 = 6$ and $524$ has digit product $5 \times 2 \times 4 = 40$. A number cannot begin with the digit $0$.
Determine the three-digit number that equals exactly five times its own digit product.