Remainder of Power of 3 Divided by 11

CAT 2024 Slot 2 · QA · Easy · Number Systems

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When 33333^{333} is divided by 1111, the remainder is

  1. A.

    1

  2. B.

    10

  3. C.

    6

  4. D.

    5

Answer

D

Explanation

By Fermat's Little Theorem or modular arithmetic: 35=243≡1(mod11)3^5 = 243 \equiv 1 \pmod{11}. Now, 333=5×66+3333 = 5 \times 66 + 3. So, 3333=(35)66×33≡166×27(mod11)≡27(mod11)≡53^{333} = (3^5)^{66} \times 3^3 \equiv 1^{66} \times 27 \pmod{11} \equiv 27 \pmod{11} \equiv 5.

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