Remainder of Exponential Power

CAT 2024 Slot 1 · QA · Easy · Number Systems

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When 1010010^{100} is divided by 7, the remainder is

  1. A.

    3

  2. B.

    6

  3. C.

    1

  4. D.

    4

Answer

D

Explanation

10≡3(mod7)10 \equiv 3 \pmod 7 10100≡3100(mod7)10^{100} \equiv 3^{100} \pmod 7

By Fermat's Little Theorem, 36≡1(mod7)3^6 \equiv 1 \pmod 7. 100=6×16+4100 = 6 \times 16 + 4 3100=(36)16×34≡1×81≡4(mod7)3^{100} = (3^6)^{16} \times 3^4 \equiv 1 \times 81 \equiv 4 \pmod 7

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