AI math solver

Run on September 28, 2026: 5 inputs on 15 models, each with this tool's own instructions.

Solve this math problem step by step. Number the steps, check the result, and end with a line that starts "Final answer:".

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What is the remainder when 7^100 is divided by 13?

1. Reduce powers of 7 modulo 13:
   7^2 = 49 ≡ 10 (mod 13).
   7^3 ≡ 10 × 7 = 70 ≡ 5 (mod 13).
   7^6 ≡ 5^2 = 25 ≡ 12 ≡ -1 (mod 13).

2. Therefore, 7^12 ≡ (-1)^2 = 1 (mod 13).

3. Write 100 = 8 × 12 + 4, so:
   7^100 ≡ (7^12)^8 × 7^4 ≡ 7^4 (mod 13).

4. Compute the remaining power:
   7^4 ≡ (7^2)^2 ≡ 10^2 = 100 ≡ 9 (mod 13).

5. Check: 100 = 13 × 7 + 9, so the remainder is indeed 9.

Final answer: 9

Step 1: Use Fermat's little theorem: since 13 is prime and 13 does not divide 7, we have 7^12 ≡ 1 (mod 13).

Step 2: Reduce the exponent: 100 = 8·12 + 4, so 7^100 = (7^12)^8 · 7^4 ≡ 7^4 (mod 13).

Step 3: Compute 7^2:
7^2 = 49 = 3·13 + 10, so 7^2 ≡ 10 (mod 13).

Step 4: Square again:
7^4 ≡ 10^2 = 100 = 7·13 + 9, so 7^4 ≡ 9 (mod 13).

Check: 9 · 7^96 inverse-style check — alternatively compute 7^100 mod 13 by repeated squaring: 7^2 ≡ 10, 7^4 ≡ 9, 7^8 ≡ 81 ≡ 3, 7^16 ≡ 9, 7^32 ≡ 3, 7^64 ≡ 9. Then 7^100 = 7^64 · 7^32 · 7^4 ≡ 9 · 3 · 9 = 243 = 18·13 + 9 ≡ 9 (mod 13). Consistent.

Final answer: 9

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