Question

Find the number of zeros at the end of \[(101)!\].

Answer 1

Hint: 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.