CAT 2023 Slot 3 Solved Paper

We know that the number of factors of these two numbers is 15 . We know that the factors of 15 are $1,3,5$, and 15.
The number of factors of $\mathrm{N}$ is $\left(\mathit{p}+1\right)\cdot \left(\mathit{q}+1\right)$ (Where, $\mathit{N}={\mathit{a}}^{\mathit{p}}\cdot {\mathit{b}}^{\mathit{q}}$, and $\mathrm{a},\mathrm{b}$ are prime numbers).
Hence, the value of $\mathit{N}$ will be least when $\left(\mathit{p}+1\right)$ and $\left(\mathit{q}+1\right)$ are as close as possible and $\mathit{a}$, and $\mathit{b}$ are the least distinct prime numbers.
Hence, $\mathit{p}+1=3\Rightarrow \mathit{p}=2$, and $\mathit{q}+1=5\Rightarrow \mathit{q}=4$, and the prime numbers $\mathit{a}$, and $\mathit{b}$ are 2 , and 3 , respectively.
Hence, the lowest value of $\mathrm{N}$ is $\mathit{N}=2^4\times 3^2=144$, and the second lowest value of $\mathrm{N}$ is $\mathit{N}=2^2\times 3^4=$ 324.
Hence, the sum is $\left(144+324\right)=468$