The USSR Olympiad Problem Book
#28. (a) Prove that \(35 \; | \; (3^{6n} - 2^{6n})\) for every positive integer \(n\).
(b) \(120 \; {\large{|}} \; \left(n^5 - 5n^3 + 4n\right)\) for every integer \(n\).
(c)* \(56,786,730 \; | \; mn(m^{60} - n^{60})\) for all integers \(m, n\).
Hint. For (a), calculate \(3^6 \pmod{5}, \; 3^{12} \pmod{5}, \; 3^{18} \pmod{5}\) and so on,
looking for a pattern. Same for \(2^6 \pmod{5}, \; 2^{12} \pmod{5}, \; 2^{18} \pmod{5}\). Do modular
arithmetic mod 5. Do the same for 7.
For (b), factor the polynomial.
For (c), factor the proposed divisor and apply Fermat's Little Theorem.