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Problem #50 · September 15, 2026

Division with remainder

Find the remainder when 220262^{2026} is divided by 55.
  1. A44
  2. B22
  3. C33
  4. D11

WORKED EXAMPLE

Solution

The remainders when powers of 22 are divided by 55 are periodic: 21→22^1 \to 2, 22→42^2 \to 4, 23=8→32^3 = 8 \to 3, 24=16→12^4 = 16 \to 1, after which the cycle repeats (cycle length 44). 2026=4⋅506+22026 = 4 \cdot 506 + 2, so 22026=(24)506⋅22≡1⋅4=4(mod5)2^{2026} = (2^4)^{506} \cdot 2^2 \equiv 1 \cdot 4 = 4 \pmod 5. The remainder is 44.
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