If a^x divides b^y and xt−yz≥0, does a^z divide b^t?

[AdrianZ]

If $$a^x|b^y$$ & $$xt-yz\geq 0$$ then $$a^z|b^t$$

This is not a homework. I found this theorem in a book without having proved it, so I wondered how it could be proved.

 

[verty]

The trick is to realize that:

if xt >= yz, then b^(yz) | b^(xt)