How many number of zeros at the end of $70!$ ?

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Solution

All that we really have to do is count the multiples of 5 that appear in 70! and count multiples of 25 twice.

The reason being 0 at the end is accounted for by 10 as a factor which can appear as
$5 \times 2$

There are enough even numbers availaible to be clubbed with multiples of 5. So we need not count number of multiples of 2.

From 1 to 70 there are 14 multiples of 5 and 2 multiples of 25.

Hence answer = 14 + 2 = 16

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