Question

How many zeros in 100! ?

Answer 1

Given number is = 100!

Exponent or power of 5 in the expansion

of 100! is $=\left[\frac{100}{5}\right]+\left[\frac{100}{5^2}\right]+\left[\frac{100}{5^3}\right]+...$

# using formula : Exponent of a point no . k

in the expansion of n! is $=\left[\frac{n}{k}\right]+\left[\frac{n}{k^2}\right]+\left[\frac{n}{k^3}\right]+...$

where [0] is the greatest integer function

$\Rightarrow$ Exponent of 5 $=\left[20\right]+\left[4\right]+\left[0.8\right]+...$

$=20+4+0...$

$=24$

similarly Exponent of 2 is $=\left[\frac{100}{2}\right]+\left[\frac{100}{4}\right]+\left[\frac{100}{8}\right]+\left[\frac{100}{16}\right]$

$+\left[\frac{100}{32}\right]+\left[\frac{100}{64}\right]+\left[\frac{100}{18}\right]+...$

$=50+25+12+6+3+1+\mathrm{0...}$

$=97$

Now prime factor of 10 are 2 & 5

$\Rightarrow (10{)}^1=2^1\times 5^1$ which makes one zero at the and

similarly $100=2^2\times 5^2\Rightarrow$ In zeroes.

$\Rightarrow 5^{24}\times 2^{24}=(10{)}^{24}\Rightarrow 24$ zeroes at the end in the expansion of 100!

$\Rightarrow Ans:24$