SMO Junior · 5. Divisibility, remainders and last digits
Last-digit cycles, congruences with clever representatives, divisibility tests from place value, and GCD/LCM tricks that remove the variable.
Find last digits of huge powers and sums using cycles.
Compute remainders with congruences and convenient (even negative) representatives.
Use and combine the tests for 2, 3, 5, 9, 11 to find missing digits.
Turn divisibility by expressions in $n$ into divisors of a fixed number.
Solve GCD/LCM conditions prime by prime and write complete short proofs.