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


Remember:
  • 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.

Q102 = 5

Q52 = 2

Q22 = 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 27 will be a factor of N!.
∴ N! can be written as 27 × 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.

Q1005 = 20

Q205 = 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.

Q103 = 3

Q33 = 1

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

∴ N! can be written as 27 × 34 × X (X is coprime to both 2 and 3)

⇒ N! = 23 × 64 × 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 22. 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 247 × X

⇒ 50! = (22)23 × 2 × X

⇒ 50! = 423 × 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 22 × 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 247 × 322 × X

⇒ 50! = (22)22 × 23 × 322 × X

⇒ 50! = 422 × 322 × 23 × X

⇒ 50! = 1222 × 23 × X

∴ Highest power of 12 in 50! is 22.

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