<s>[INST] Prove divisibility using linear congruences I need to prove that:
$$10|(53^{53} - 33^{33})$$
I can and should only use linear congruences ($a \equiv b \mod n$) - how can I do this?
 [/INST] $$53^{53}-33^{33}\equiv 3^{53}- 3^{33}\pmod {10}$$
Now, $$3^4=81\equiv 1\pmod 4\implies 3^{53}- 3^{33}\equiv 3^1-3^1\pmod {10}\equiv0$$ as $33\equiv1\pmod 4, 53\equiv1\pmod 4$
Alternatively, $$3^{53}- 3^{33}\equiv 3^{33}(3^{20}-1)\pmod {10}$$
Now, $3^4\equiv1\pmod 4\implies 3^{20}=(3^4)^5\equiv1^5\pmod {10}\equiv1$
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