# Exonent of 12 in100!(factorial)

1
by krishnaMalhotra
Answer is 48. Find the exponents of 2² and 3 pick the smaller one.

2014-11-30T16:13:49+05:30

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Exponent of a prime number p  in a number N  is given as  a  if
N = p^a * M,  where  M is the product of all remaining factors.

If p, q, r are prime numbers or relatively prime, then

Then exponent of (pq) in N is the smaller of the two numbers a and b.

Exponent of a number p in N!  is given by the expression :

This is so because,  there are N/p multiples of p in N like p, 2p, 3p etc., then for each p^2, there is one extra p.  For p^3 , there is one more extra  p  as compared to p^2.   Hence we get the above formula.
12 = 2^2 * 3.  So we find exponents of 2 and 3 in 100 !.    Then we find exponent of 12.

Hence exponent of 12 in 100! is 48.