Prove that if $a\mid(2b^2+5c^2)$, then $a\nmid(3b^2-2c^2)$ or $a\le{19c^2}$.

Question

Prove that if $a\mid(2b^2+5c^2)$, then $a\nmid(3b^2-2c^2)$ or $a\le{19c^2}$.

I have proved that $a\le{19c^2}$ however I'm not sure how to prove that $a\nmid(3b^2-2c^2)$.

How should I go about this?

Thanks!

Answer 1

Either you have $a\not|(3b^2-2c^2)$ in which case you are immediately done, or $a|(3b^2-2c^2)$, in which case $a$ divides $$ \Big[3\times(2b^2+5c^2)-2\times(3b^2-2c^2)\Big]=19c^2 $$ so in particular $a\leq 19c^2$.