Question

Find the number of zeros at the end of $(101)!$.

Answer 1

We write the terms that are multiplied while opening the factorial. Since each zero at the end is made up of a 5 multiplied by 2, we count the number of terms that are divisible by 5.

• Factorial of any number ‘n’ is given by$n! = n \times (n - 1) \times (n - 2) \times ..... \times 3 \times 2 \times 1$

Complete step-by-step answer:
Since we know a factorial of any number is given by $n! = n \times (n - 1) \times (n - 2) \times ..... \times 3 \times 2 \times 1$
Here $n = 101$
$\Rightarrow 101! = 101 \times (101 - 1) \times (101 - 2) \times ..... \times 3 \times 2 \times 1$
$\Rightarrow 101! = 101 \times (100) \times (99) \times ..... \times 3 \times 2 \times 1$
So, terms that are multiplied in the factorial are $101,100,99,98,97,.......5,4,3,2,1$
We know $10 = 2 \times 5$
Since there are 50 even numbers between 1 and 101
Therefore we find the terms in the factorial that are divisible by 5
Numbers that are divisible by 5 are: $5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100$
$\Rightarrow$There are 20 terms divisible by 5
$\Rightarrow$There are 20 zeroes at the end of$101!$............… (1)
But there are terms that are divisible twice by 5 i.e. they are divisible by $5 \times 5 = 25$
Numbers that are divisible by 25 are: $25,50,75,100$
$\Rightarrow$There are 4 terms divisible by 25
$\Rightarrow$There are 4 more zeroes at the end of $101!$..........… (2)
So, total number of zeros after $101!$ is given by adding the number of zeros from equation (1) and (2)
$\Rightarrow$Number of zeros at the end of $101! = 20 + 4$
$\Rightarrow$Number of zeros at the end of $101! = 24$

$\therefore$Number of zeros at the end of $101!$ is 24.

Note: Students might try to solve for the value of $101!$ by multiplying all the values of factorial given by $101! = 101 \times (100) \times (99) \times ..... \times 3 \times 2 \times 1$. But since there are 101 numbers to be multiplied with each other, this will be a very long and complex calculation. Students are advised not to proceed in this manner. Also, many students only write 20 zeroes as they don’t see the terms having ${5^2}$ in them.