Video narration
Trying to calculate three to the hundredth power directly feels absurd. But if you only need its remainder modulo ten-its last digit-you can think like a clock. Powers of three tick through four positions: three, nine, seven, one, then three again. So every four steps, the remainder returns to one. Coprime means sharing no factor greater than one. Among numbers up to ten, one, three, seven, and nine are coprime to ten, so φ(n), the count of such numbers, is four. Euler's theorem says that when the base and modulus are coprime, raising the base to φ(n) gives remainder one. The calculation flips: you never find the huge power. One hundred is twenty-five groups of four, so three to the hundredth power modulo ten is one. This clock keeps working because each cycle brings you back to one. Remember: check coprime, find φ(n), reduce the exponent.