Question

How many zeroes are there in product of first 100 natural number

Answer 1

$no \: of \: zeroes \\ = greatest \: integer \: function \: of \\ \: ( \frac{100}{5} ) + ( \frac{100}{25} ) = 20 + 4 = 24$

Answer 2

Given Question is to find number of zeroes in product of 100 natural numbers = 100!.

We know that each pair of 2 and 5 will give one zero.

(i)

Factors of 10 = 2 and 5.

As we know that there are so many 2 from 1 to 100, so, the numbers of zeroes are limited by number of 5 factors in it.

(ii)

Number having at least one 5 as factor in 1-100:

⇒ 100/5

⇒ 20{5,10,15,20,...100}

(iii)

Number having at least two 5's as factor in 1-100:

⇒ 100/5^2

⇒ 100/25

⇒ 4. {25(5 * 5),50(5 * 5),75(5 * 5),100(5 * 5)}

So, the total count of fives = 20 + 4 = 24.

Therefore, the number of zeroes = 24.

Hope this helps!