The Brainliest Answer!

This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
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
       N = p^a * q^b * r^c * ...

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 :

E_p(N!) = \frac{N}{p}+\frac{N}{p^2}+\frac{N}{p^3}+... +\frac{N}{p^s}\\\\ where,\ s\ is\ maximum\ integer\ such\ that\ p^s<=N\\

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.

E_2(100!)=\frac{100}{2}+\frac{100}{4}+\frac{100}{8}+\frac{100}{16}+\frac{100}{32}+\frac{100}{64} = 97\\\\E_3(100!)=\frac{100}{3}+\frac{100}{9}+\frac{100}{27}+\frac{100}{81}=48\\\\100!=2^{97}*3^{48}*M,\ \ M=product\ of\ other\ factors.\\\\ .\ \ \ =2*4^{48}*3^48*M=2*12^{48}*M\\

Hence exponent of 12 in 100! is 48.

2 5 2
Actual answer is 48.
You may delete my comments.
click on thank you and select best answer.
Its a good explanation.....