Let $\mathrm{f}, \mathrm{g} : \mathrm{\mathbb{N}}-\left\{1\right\}\to \mathrm{\mathbb{N}}$ be functions defined by $\mathrm{f}\left(\mathrm{a}\right)=\mathrm{\alpha }$, where $\mathrm{\alpha }$ is the maximum of the powers of those primes p such that ${\mathrm{p}}^{\mathrm{\alpha }}$ divides a, and $\mathrm{g}\left(\mathrm{a}\right)=\mathrm{a}+1$, for all $\mathrm{a}\in \mathrm{\mathbb{N}}-\left\{1\right\}$. Then, the function f + g is

$\mathrm{f} : \mathrm{N}-\left\{1\right\}\to \mathrm{N} \mathrm{f}\left(\mathrm{a}\right)=\mathrm{\alpha }$

Where $\mathrm{\alpha }$ is max of powers of prime P such that ${\mathrm{p}}^{\mathrm{\alpha }}$ divides a. Also g(a) = a + 1

$\begin{array}{l}\therefore \mathrm{f}\left(2\right)=1\mathrm{g}\left(2\right)=3\\ \begin{array}{ll}\mathrm{f}\left(3\right)=1& \mathrm{g}\left(3\right)=4\\ \mathrm{f}\left(4\right)=2& \mathrm{g}\left(4\right)=5\\ \mathrm{f}\left(5\right)=1& \mathrm{g}\left(5\right)=6\end{array}\\ \Rightarrow \mathrm{f}\left(2\right)+\mathrm{g}\left(2\right)=4\\ \left(\mathrm{f}\right(3)+\mathrm{g}(3\left)\right)=5\\ \mathrm{f}\left(4\right)+\mathrm{g}\left(4\right)=7\\ \mathrm{f}\left(5\right)+\mathrm{g}\left(5\right)=7\\ \therefore \text{ Many one }\mathrm{f}\left(\mathrm{x}\right)+\mathrm{g}\left(\mathrm{x}\right)\text{ does not cotain }1\\ \Rightarrow \text{ into function }\end{array}$

$\therefore$  Ans. (D) [neither one-one nor onto ]