# Concept: Factorial

**INTRODUCTION**

Factorial, in mathematics, is the product of all positive integers less than or equal to a given positive integer and is denoted by that integer and an exclamation point.

Thus, factorial four is written 4!, which means 1 × 2 × 3 × 4

Hence, n! = 1 × 2 × 3 × ... × (n-1) × n

- n! = n × (n - 1)! = n × (n - 1) × (n - 2)!
- 0! = 1
- 1! = 1
- 2! = 2
- 3! = 6
- 4! = 24
- 5! = 120
- 6! = 720

**HIGHEST POWER OF P (PRIME) IN N!**

If 'p' is a prime number, then highest power of 'p' in N! is sum of all quotients when N is successively divided by p.

**Example**: Find the highest power of 2 in 10!.

**Solution:**

To find the highest power of 2 in 10!, we successively divide 10 by 2 and then add all the quotients.

$\mathrm{Q}\left[\frac{10}{2}\right]$ = 5

$\mathrm{Q}\left[\frac{5}{2}\right]$ = 2

$\mathrm{Q}\left[\frac{2}{2}\right]$ = 1

(We stop the division when quotient is less than the divisor)

∴ Highest power of 2 in 10! = 5 + 2 + 1 = 7

**Note**: Since, highest power of 2 in 10! is 7, it means 2^{7} will be a factor of N!.

∴ N! can be written as 2^{7} × X

where X is a factor of N! which is coprime to 2.

**Example**: Find the highest power of 5 in 100!.

**Solution:**

To find the highest power of 5 in 100!, we successively divide 100 by 5 and then add all the quotients.

$\mathrm{Q}\left[\frac{100}{5}\right]$ = 20

$\mathrm{Q}\left[\frac{20}{5}\right]$ = 4

∴ Highest power of 5 in 100! = 20 + 4 = 24

**HIGHEST POWER OF A COMPOSITE NUMBER IN N!**

To find highest power of a composite number in N!, we first prime factorise the given composite number and then find the highest powers of prime numbers.

**Example**: Find the highest power of 6 in 10!.

**Solution:**

To find the highest power of 6 in 10!, we prime factorise 6 as 2 × 3. Now we will find the highest power of 2 and 3 in 10!

Highest power of 2 in 10! is 7 (obtained in previous example).

To find the highest power of 3 in 10!, we successively divide 10 by 3 and then add all the quotients.

$\mathrm{Q}\left[\frac{10}{3}\right]$ = 3

$\mathrm{Q}\left[\frac{3}{3}\right]$ = 1

∴ Highest power of 3 in 10! = 3 + 1 = 4

∴ N! can be written as 2^{7} × 3^{4} × X (X is coprime to both 2 and 3)

⇒ N! = 2^{3} × 6^{4} × X

∴ Highest power of 6 in 10! is 4.

**Example**: Find the highest power of 4 in 50!.

**Solution:**

To find the highest power of 4 in 5!, we prime factorise 4 as 2^{2}. Now we will find the highest power of 2 in 50!

To find the highest power of 2 in 50!, we successively divide 50 by 2 and then add all the quotients.

∴ Highest power of 2 in 50! = 25 + 12 + 6 + 3 + 1 = 47

∴ 50! can be written as 2^{47} × X

⇒ 50! = (2^{2})^{23} × 2 × X

⇒ 50! = 4^{23} × 2 × X

∴ Highest power of 4 in 50! is 23.

**Example**: Find the highest power of 12 in 50!.

**Solution:**

To find the highest power of 12 in 50!, we prime factorise 12 as 2^{2} × 3. Now we will find the highest power of 2 and 3 in 50!

Highest power of 2 in 50! is 47 (obtained in previous example).

To find the highest power of 3 in 50!, we successively divide 50 by 3 and then add all the quotients.

∴ Highest power of 3 in 50! = 16 + 5 + 1 = 22

∴ 50! can be written as 2^{47} × 3^{22} × X

⇒ 50! = (2^{2})^{22} × 2^{3} × 3^{22} × X

⇒ 50! = 4^{22} × 3^{22} × 2^{3} × X

⇒ 50! = 12^{22} × 2^{3} × X

∴ Highest power of 12 in 50! is 22.