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The Last Digit of a Colossal Power

Issue 21 Warm-up Published 30 Sep 2026 By The Chalkboard Desk

Tier Warm-up — The Last Digit of a Colossal Power

The problem

What is the remainder when is divided by ? Give the answer as an integer between and .
  • You may not compute — it has digits.
  • Only remainders are needed at each step.

Take a minute

Reveal hint 1
Reducing the base modulo is always allowed: .
Reveal hint 2
Powers of modulo repeat with period : .

Solution

Answer.

Derivation

  1. Reduce the base.

  2. Find the cycle length of powers of .

  3. Reduce the exponent modulo the cycle length.

  4. Justify the shortcut.

  5. State the answer in the required range.