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Answer: 6 zeros


Multiplying by $10$, or multiples/powers of $10$, makes zeros, so find powers of $10$
    $3^4\times 4^5\times 5^6$
$=3^4\times 2^{10}\times 5^6$
$=3^4\times 2^4\times 2^6 \times 5^6$
$=\underbrace{6^4}_{\text{no zeros}}\times\underbrace{10^6}_{\text{six zeros}}$


This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.