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.

Hint: $\,\gcd(a^2\!+\!1,a^4\!+\!1) = \gcd(a^2\!+\!1,(a^2\!+\!1)(a^2\!-\!1) + 2) = \gcd(a^2\!+\!1,2)$
