Find the number f divisors of 720. How many of these are even? Also find the sum of divisors.

Explanation

We have
`720=2^(4)xx3^(2)xx5^(1)`
Now, number of divisors of 720 is equivalent to selection of zero or more integer from (2,2,2,2),(3,3),(5).
The number of ways of selection is `5xx3xx2=30`. This also inclueds 1 and 720.
Now, in even divisor factor 2 should be there. Then from set of 2, we must have nonempty selection. Hence, the number of even divisors is `4xx3xx2=24`. Also, the number of divisor of 720 can be seen as the number of different terms in the expansion of `(2^(0)+2^(1)+2^(2)+2^(3)+2^(4))xx(3^(0)+3^(1)+3^(2))xx(5^(0)+5^(1))`
which also gives sum of divisors.
Hence, sum of divisors is
`(2^(0)+2^(1)+2^(2)+2^(3)+2^(4))xx(3^(0)+3^(1)+3^(2))xx(5^(0)+5^(1))`
`(2^(5)-1)/(2-1)xx(3^(3)-1)/(3-1)xx(5^(2)-1)/(5-1)=2418`