Introduction:-
There are two ways to represent a factorial number:-
N! = N × (N-1) × (N-2) × (N-3) × ……….. × 1
0!= 1
1!= 1
2!= 2
3!= 6
4!= 24
5!= 120
6!= 720
Highest power of a number in N!:-
method (1):-
Highest power of prime number p that divides n! exactly i.e. without leaving any remainder is given by:-
where [a] represents greatest integer less than or equal to a.
Example:- Find the highest power of 3 in 100!
Solution:-
= 33 + 11 + 3 + 1
= 48 Answer
method (2):-
Quotient form → Successive division
(Q). Find the highest power of 6 in 150!
Solution:- First, since 6 is a composite number so convert it into its prime factors.
6 = 2 × 3
now
So highest power of 2 in 150! = 146
& highest power of 3 in 150! = 72
∴ Highest power of 6 in 150! = minimum of above two values = 72 Answer