If $a, b$ are positive integers, does $\;b\mid(a^2 + 1)\implies b\mid (a^4 + 1)\quad?$ If $a, b$ are positive integers, does $\;b\mid(a^2 + 1)\implies b\mid (a^4 + 1)\;$?
Explain if this is true or not. If no, give a counterexample.

Suppose $b = 5, a = 3$:
$$5\mid (3^2 + 1)\;\;\text{but}\;\;5 \not\mid (3^4 + 1)$$
Or, simply choose $a = 2, b= 5$ and again, $$5\mid (2^2 + 1)\;\;\text{but}\;\;5 \not\mid (2^4 + 1)$$

What is true is that  $b\mid (a^2 + 1) \implies b\mid(a^2 + 1)^2$, 
but note that $$\begin{align}\;(a^2 + 1)^2 &= (a^4 + 2a^2 + 1) \\ \\ &= \left[(a^4 + 1) + 2a^2\right]\\ \\ &\neq (a^4 + 1)\end{align}$$
