Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$ Let $a,b \in {\mathbb{Z_+}}$ such that $$a|b^2,  b^3|a^4, a^5|b^6, b^7|a^8 \cdots$$
Prove $a=b$

consider the prime factorisations of $a = \prod_p p^{\nu_p(a)}$ and $b = \prod_p p^{\nu_p(b)}$. Your assumptions yield


*

*$\nu_p(a) \le \nu_p(b^2) = 2\nu_p(b)$ for each $p$

*$3\nu_p(b) \le 4\nu_p(a)$

*$\ldots$

*$(4n+1)\nu_p(a) \le (4n+2)\nu_p(b)$ for each $p$, $n$

*$(4n+3)\nu_p(b) \le (4n + 4)\nu_p(a)$, each $p$, $n$


So we have for each $p$, $n$
$$
  \frac{4n + 3}{4n+4} \cdot \nu_p(b) \le \nu_p(a) \le \frac{4n+2}{4n+1}\nu_p(b)
$$
letting $n\to \infty$ yields $\nu_p(a) = \nu_p(b)$ for each $p$, so $a = b$.
AB,
