How many positive factors does

Question

62x(63^3+63^2+63+1)+1 have?

(with solution pls... thanks...)

Answer 1

63^3 + 63^2 + 63 + 1
= 63^2(63 + 1) + (63+1)
= (63^2 + 1)(63+1)
= 3970 x 64

62x(63^3+63^2+63+1)+1
= 62 x 3970 x 64 + 1
= 15752961
= 2401x6561
= 2401 x 3^8
= 7x343x3^8
= 7^4 x 3^8

So now we need to take different combinations of factors, e.g. 7x7 x 3x3x3 would be a factor

the four factors of 7 can be taken in 5 ways,
that is, take none, take one, take 2, take 3 or take 4 of them
in the same way, the eight 3's can be taken in 9 ways.
So the total number of subsets of the above is
5x9 or 45
BUT, that would include taking neither the 7 nor the 3
so we subtract 1

number of positive factors is 44