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.