UPSC CDS 2 2022 Math Paper

We know Total number of factors of $x^{p} y^{q}$ is $(p+1)(q+1)$Number of odd factors of $x^{p} y^{q} z^{r}$ if $x$ is even and $y$ and $z$ are odd then the factors are$1 \times (q+1)(r+1)$ We have,$\Rightarrow N=12^6 \times 3^8 \times 5^3$$\Rightarrow N=2^6 \times 2^6 \times 3^6 \times 3^8 \times 5^3$$\Rightarrow N=2^{12} \times 3^{14} \times 5^3$Total number of factors:- $(12+1) \times (14+1) \times (3+1)$Total number of factors:- $13 \times 15 \times 4$Total number of factors $=780$Statement I: The number of odd factors of $N$ is 60 .Number of odd factors $=1 \times (14+1) \times (3+1)$Number of odd factors $=1 \times 15 \times 4$A number of odd factors $=60$.Hence, Statement I is correct.Statement II: The number of even factors of $N$ is 720 .Number of even factors $=$ Total number of factors $-$ Odd factorsNumber of even factors $=780-60$Number of even factors $=720$Hence, Statement II is correct.$\therefore$ Both statements are correct.