A number theory problem by Pankhuri Agarwal

Prime factorization of $N$ is $N={ 2 }^{ 6 }\cdot { 3 }^{ 7 }\cdot { 5 }^{ 2 }$ .

The number of all factors is $\left( 6+1 \right) \cdot \left( 7+1 \right) \cdot \left( 2+1 \right) =168$ .

To calculate the number of even factors we can calculate the number of odd factors and subtract it from the number of all factors. In order to calculate it we can just ignore the number $2$ in prime factorization of $N$ .

So the number of odd factors is equal to number of factors of ${ N }_{ 1 }={ 3 }^{ 7 }{ \cdot 5 }^{ 2 }$ which is equal to $\left( 7+1 \right) \cdot \left( 2+1 \right) =24$ .

Therefore the number of even factors is $168-24=144$ .

The complete prime factorization of $N$ is $3^{7} \times 5^{2} \times 2^{6}$ .

The total number of even factors is therefore

$(7+1) \times (2+1) \times 6$

$\boxed {144}$

Prime factorisation is $2^6 \cdot 3^7 \cdot 5^2.$

Number of factors is the number of of combinations of $2^a \cdot 3^b \cdot 5^c$ such that $0 <= a <= 6$ , $0 <= b <= 7$ and $0 <= c <= 2$ (and of course, $a$ , $b$ and $c$ are integers). To get the number of even factors, we just ignore the case $a = 0$ . There are thus $6 \cdot 8 \cdot 3 = \boxed{144}$ even factors.

(2^2 * 3)^3 * 3^4 * 5^2

N = 2^6 * 3^7 * 5^2

even factors = (6)(8)(3) = 144

the total factors of N = 168 the total odd factors = 24 the total even factors = 168 - 24 = 144 booommmmmmmmmmmm