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**Haven** I am having troubles with this problem:

For any n find all pairs $\displaystyle a,b \in \mathbb{N}$ such that $\displaystyle lcm(a,b) = n$

All I have managed to prove is that:

$\displaystyle \frac{a}{gcd(a,b)} | n$ and $\displaystyle \frac{b}{gcd(a,b)} | n$

Or if $\displaystyle a=p_1^{a_1} \dots p_t^{a_t}, b=p_1^{b_1} \dots p_t^{b_t}, gcd(a,b) = p_1^{d_1} \dots p_t^{d_t}$ and $\displaystyle n = p_1^{n_1} \dots p_t^{n_t}$

Then for all $\displaystyle i \in \{1, \dots ,t\}$ $\displaystyle a_i + b_i = c_i + d_i$