Consider positive distinct integers $a_1,a_2,\dots ,a_n$. For each positive integer $N$, define the set $$S_N=\{Na_1,Na_2,\dots ,Na_n\}$$
My question is, for any original collection $a_i$, does there exist an $N$ such that $\tau(Na_i)$ is distinct for each $i$? I.e. $\tau$ is injective over $S_N$. Here, $\tau$ denotes the number of divisors function.
For example, if we started with $\{2,3,5,8\}$, we could take $N=45$. I think we might be able to construct $N$ based on the prime factors of $a_i$, or use the Chinese Remainder Theorem somehow.
$$a_j = \prod_{k=1}^K p_k^{e_{j,k}}, \qquad r = 1+\sup_{j,k} e_{j,k},\qquad b = \prod_{k=1}^K p_k^{rk}, \qquad c=\prod_{k=1}^K p_k$$ then $$\tau(a_jbc^x)-\tau(a_ibc^x)=\prod_{k=1}^K (e_{j,k}+rk+x+1)-\prod_{k=1}^K (e_{i,k}+rk+x+1)$$ is a non-zero polynomial in $x$ with only finitely many roots, thus for $x$ large enough $$\tau(a_j b c^x)\ne \tau(a_i b c^x)$$