31.29. a) Provide an example of numbers $m \neq n$ for which $\varphi(n)=\varphi(m)$.

b) Prove that if $m \neq n$, then $n \varphi(n) \neq m \varphi(m)$.

c) Provide an example of numbers $m \neq n$ for which $n \sigma(n) = m \sigma(m)$.
31.29. a) Answer: 5 and 8.

b) Let $n=p_{1}^{a_{1}} \ldots p_{k}^{a_{k}}$ and $p_{1}>p_{2}>\ldots>p_{k}$. Then

$$
n \varphi(n)=\prod_{i=1}^{k}\left(p_{i}^{2 a_{i}-1}\left(p_{i}-1\right)\right)
$$

and the largest prime number that divides $n \varphi(n)$ is $p_{1}$. The maximum power of the number $p_{1}$ that divides $n \varphi(n)$ is $2 a_{1}-1$. Therefore, if the number $n \varphi(n)$ is known, then the numbers $p_{1}$ and $a_{1}$ are known. Let $n=p_{1}^{a_{1}} n_{2}$. Then $n_{2} \varphi\left(n_{2}\right)=\frac{n \varphi(n)}{p_{1}^{2 a_{1}-1}\left(p_{1}-1\right)}$, so the numbers $p_{2}$ and $a_{2}$ are known, and so on.

c) Answer: 12 and 14.