the prime factorization of 10010 what is the exponent of the

the prime factorization of (100!)^10 what is the exponent of the prime 31?

Solution

31 is a prime number hence it has no factors

it is enough to know the multiples of 31

100!=1.2.3........................31...................62....................93.........100

  =31.62.93.k where k is not a multiple of 31

now 100!^10=(31.62.93.k)^10

=(31.2.31.3.31)^10.k^10

=[31^3.(2.3.)]^10.k^10

=(31^3)^10.(6k)^10

=31^30.(6k)^10

there fore the exponent of the prime 31 is 30