6

We need to write $3^4\times 4^5\times 5^6$ in the form $a\times 10^n$ where $a$ is not a multiple of 10.

$3^4\times 4^5\times 5^6=3^4\times 2^{10}\times 5^6=3^4\times 2^4\times 2^6 \times 5^6=3^4\times 2^4\times 10^6.$

Hence the number ends in six zeros.

*This problem is taken from the UKMT Mathematical Challenges.*