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Problem #45 · September 10, 2026

Powers and roots

Find the last digit of the sum 22026+320262^{2026} + 3^{2026}.
  1. A44
  2. B99
  3. C33
  4. D55
  5. E77

WORKED EXAMPLE

Solution

The last digits of powers of 2 repeat in the cycle 2,4,8,62, 4, 8, 6, and those of powers of 3 repeat in the cycle 3,9,7,13, 9, 7, 1 (cycle length 4). Since 2026=4⋅506+22026 = 4 \cdot 506 + 2, the number 220262^{2026} ends in 4, and the number 320263^{2026} ends in 9. 4+9=134 + 9 = 13, so the last digit of the sum is 3.
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