Question

If N = ${12}^3\times 3^4\times 5^2$ find the total number of even factors of N.

Answer 1

The factorised form of N is
$(2^2\times 3^1)\times 3^4\times 5^2\Rightarrow 2^6\times 3^7\times 5^2$
Hence the total number of factors of N is (6 + 1) (7 + 1) (2 + 1) = 7 x 8 x 3 = 168
Some of these are odd multiples and some are even The odd multiples are formed only with the combination of 3s and 5s
So the total number of odd factors is (7 + 1) (2 + 1) = 24
Therefore the number of even factors = 168 - 24 = 144