Quite a simple problem, two-liner with LTE... First notice that $\frac{a_{s}-a_{t}}{s-t}=a^t \cdot \frac{a^{s-t}-1}{(a-1)(s-t)}$(for $s>t$, other case has the same proof). Now, as for some prime $p$ dividing $s-t$, hence $a-1$, as we know $a \equiv 1 \pmod{p}$, we get $v_p(a^{s-t}-1)=v_p(a-1)+v_p(s-t)$. Now for all the primes $q$ that divide $a-1$ but not $s-t$ (basically all other primes in the denominator) clearly divide $a^{s-t}-1$, as $v_q(a^{s-t}-1)=v_q(a-1)+v_q(s-t)$, hence we're done. Btw what's FYROM supposed to mean?

FYROM = North Macedonia