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चरण द्वारा कदम

1. 7^{222} \bmod 11

   Never compute the huge power. Reduce after every multiplication (repeated squaring).

2. 7 \equiv 7 \pmod{11}

   Reduce the base first.

3. 222 = 11011110_2

   Write the exponent in binary: 8 squarings at most.

4. 7^2 \equiv 5 \pmod{11}

   Square the running power.

5. r \leftarrow r \cdot 5 \equiv 5 \pmod{11}

   This bit is 1: multiply the result by the current power.

6. 5^2 \equiv 3 \pmod{11}

   Square the running power.

7. r \leftarrow r \cdot 3 \equiv 4 \pmod{11}

   This bit is 1: multiply the result by the current power.

8. 3^2 \equiv 9 \pmod{11}

   Square the running power.

9. r \leftarrow r \cdot 9 \equiv 3 \pmod{11}

   This bit is 1: multiply the result by the current power.

10. 9^2 \equiv 4 \pmod{11}

    Square the running power.

11. r \leftarrow r \cdot 4 \equiv 1 \pmod{11}

    This bit is 1: multiply the result by the current power.

12. 4^2 \equiv 5 \pmod{11}

    Square the running power.

13. 5^2 \equiv 3 \pmod{11}

    Square the running power.

14. r \leftarrow r \cdot 3 \equiv 3 \pmod{11}

    This bit is 1: multiply the result by the current power.

15. 3^2 \equiv 9 \pmod{11}

    Square the running power.

16. r \leftarrow r \cdot 9 \equiv 5 \pmod{11}

    This bit is 1: multiply the result by the current power.

17. 7^{222} \equiv 5 \pmod{11}

    Done.

जवाब दिखाएँ

7^{222} \bmod 11 = 5