Fermat's little theorem and Euler's theorem

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1. 2^{100} \bmod 13

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

2. 2 \equiv 2 \pmod{13}

   Reduce the base first.

3. 100 = 1100100_2

   Write the exponent in binary: 7 squarings at most.

4. 2^2 \equiv 4 \pmod{13}

   Square the running power.

5. 4^2 \equiv 3 \pmod{13}

   Square the running power.

6. r \leftarrow r \cdot 3 \equiv 3 \pmod{13}

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

7. 3^2 \equiv 9 \pmod{13}

   Square the running power.

8. 9^2 \equiv 3 \pmod{13}

   Square the running power.

9. 3^2 \equiv 9 \pmod{13}

   Square the running power.

10. r \leftarrow r \cdot 9 \equiv 1 \pmod{13}

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

11. 9^2 \equiv 3 \pmod{13}

    Square the running power.

12. r \leftarrow r \cdot 3 \equiv 3 \pmod{13}

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

13. 2^{100} \equiv 3 \pmod{13}

    Done.