What is the number of Odd divisors of 20! ?
We have to find out Number of Odd factors of 20!. So Let us start by Expressing 20! in terms of Prime factors. Since
N = 20 X 19 X18...................1 = 2^m + 3^n + 5^p + 7^q. Our First goal is to find the Highest Power of 2, 3, 5, 7, 11, 17, 19 which divide 20!.
Highest Power of 2 Which will divide 20! is = 20/2 + 20/4 + 20/8 + 20/16
= 10 + 5 + 2 + 1 = 18. Remember only Quotients need to be considered
the highest Power of 3, 5, 7,11,13,17,19 is 8 ,4,2,1,1,1,1 respectively.
express 20! as
20! = 2^18 + 3^8 + 5^4 + 7^2 + 11^1 + 13^1 + 17^1 +19^1.
Total no of odd Divisors is (8+1) x ( 4+1) x ( 2+1) X (1+1) x(1+1) x(1+1)x(1+1) = 9 x 5x3x16
why the + 1 when multiplying:
Because the 3 ^ 0 should also be counted in determining the number of factors. Similarly, the powers for other primes have a +1.
why they are multiplied:
To determine the total no. of distinct factors (ways they can be combined) that can be formed from the given set of prime factors
clipped from: www.urch.com