A remainder of a large power: every AI model's reply, tested

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?

How the AI math solver's test inputs are scored, with every model's results on all five.

1. GPT-6 Luna

   Passed: Final answer 9: right.

   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

   341 tokens in, 342 out (122 of them reasoning) · 4.0 s · $0.0001 · 1 message on Pro · answered by openai/gpt-6-luna via OpenAI · September 28, 2026

2. GLM 5.3 Flash

   Passed: Final answer 9: right.

   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

   349 tokens in, 366 out (22 of them reasoning) · 3.4 s · $0.0002 · 1 message on Pro · answered by z-ai/glm-5.3-flash via Friendli · September 28, 2026

3. Claude Haiku 5.5

   Passed: Final answer 9: right.

   Step 1: Find the pattern of powers of 7 modulo 13.
   7^1 = 7 (mod 13), so 7^1 ≡ 7.
   
   Step 2: Compute the next powers.
   7^2 = 49. Since 49 = 3·13 + 10, 7^2 ≡ 10 (mod 13).
   
   Step 3: Continue.
   7^3 ≡ 10·7 = 70 = 5·13 + 5, so 7^3 ≡ 5 (mod 13).
   
   Step 4: Continue.
   7^4 ≡ 5·7 = 35 = 2·13 + 9, so 7^4 ≡ 9 (mod 13).
   
   Step 5: Continue.
   7^5 ≡ 9·7 = 63 = 4·13 + 11, so 7^5 ≡ 11 (mod 13).
   
   Step 6: Continue.
   7^6 ≡ 11·7 = 77 = 5·13 + 12, so 7^6 ≡ 12 (mod 13).
   
   Step 7: Notice 12 ≡ -1 (mod 13), so 7^6 ≡ -1 and 7^12 ≡ (-1)^2 = 1 (mod 13).
   The order of 7 modulo 13 divides 12, and we have 7^6 ≡ -1 ≠ 1, so the order is 12 (we also checked the smaller candidates 1, 2, 3, 4, 6 are not valid since 7^6 ≠ 1 and 7^4 ≠ 1, 7^3 ≠ 1, 7^2 ≠ 1, and the order must divide 12 and cannot be 6, so it is 4 or 12 or 3 or 2 or 1; 7^4 ≡ 9 ≠ 1 rules out 1, 2, 4, and 7^3 ≡ 5 ≠ 1 rules out 3, so the order is 12).
   
   Step 8: Reduce the exponent.
   100 = 8·12 + 4, so 7^100 = (7^12)^8 · 7^4 ≡ 1^8 · 7^4 ≡ 7^4 (mod 13).
   
   Step 9: Use the value from Step 4.
   7^4 ≡ 9 (mod 13).
   
   Step 10: Check.
   Direct check by another route: 7^4 = 2401. Dividing 2401 by 13: 13·184 = 2392, and 2401 − 2392 = 9. This matches 9.
   Also check 7^100 ≡ 7^4 via Fermat's little theorem: 7^12 ≡ 1 (mod 13), and 100 mod 12 = 4, which agrees with Step 8.
   
   Final answer: 9

   447 tokens in, 895 out · 4.0 s · $0.0005 · 1 message on Pro · answered by anthropic/claude-haiku-5.5 via Anthropic · October 7, 2026

4. DeepSeek V4.1 Flash

   Passed: Final answer 9: right.

   1. We want the remainder of 7^100 when divided by 13, which means compute 7^100 mod 13.
   
   2. Since 7 and 13 are coprime, Fermat's Little Theorem gives 7^12 = 1 mod 13.
   
   3. Divide the exponent 100 by 12:
   100 = 12 * 8 + 4, so 100 mod 12 = 4.
   
   4. Therefore,
   7^100 = 7^(12*8 + 4) = (7^12)^8 * 7^4.
   Since 7^12 = 1 mod 13, this becomes
   7^100 = 1^8 * 7^4 = 7^4 mod 13.
   
   5. Now compute 7^4 mod 13:
   7^2 = 49, and 49 mod 13 = 10.
   So 7^4 = (7^2)^2 = 10^2 = 100.
   Then 100 mod 13 = 100 - 91 = 9.
   
   6. Check another way:
   7^6 = 117649, and 117649 mod 13 = 12 = -1.
   Then 7^100 = (7^6)^16 * 7^4 = (-1)^16 * 9 = 9 mod 13.
   This agrees.
   
   Final answer: 9

   367 tokens in, 536 out (244 of them reasoning) · 1.1 s · $0.0007 · 1 message on Pro · answered by deepseek/deepseek-v4.1-flash via Together · October 8, 2026

5. GLM 5.3

   Passed: Final answer 9: right.

   Step 1: Use Fermat's little theorem: since 13 is prime and does not divide 7, we have 7^12 ≡ 1 (mod 13).
   
   Step 2: Divide the exponent by 12 to reduce it:
   100 = 12 · 8 + 4, so 100 mod 12 = 4.
   
   Step 3: Therefore 7^100 ≡ 7^4 (mod 13).
   This follows because 7^100 = (7^12)^8 · 7^4 ≡ 1^8 · 7^4 = 7^4.
   
   Step 4: Compute 7^4:
   7^2 = 49, and 49 mod 13 = 49 - 39 = 10.
   Then 7^4 ≡ 10^2 = 100 (mod 13).
   100 = 13 · 7 + 9, so 7^4 ≡ 9 (mod 13).
   
   Step 5: Check the result directly by computing 7^4 = 2401 and dividing:
   13 · 184 = 2392, and 2401 - 2392 = 9. This confirms the remainder is 9.
   
   Final answer: 9

   349 tokens in, 313 out (22 of them reasoning) · 1.6 s · $0.0014 · 1 message on Pro · answered by z-ai/glm-5.3 via Friendli · September 28, 2026

6. Gemini 3.8 Flash

   Passed: Final answer 9: right.

   Step 1: State the goal using modular arithmetic.
   We want to find the remainder of 7^100 when divided by 13, which means finding a value r such that:
   7^100 = r (mod 13), with 0 <= r < 13.
   
   Step 2: Apply Fermat's Little Theorem.
   Fermat's Little Theorem states that if p is a prime number and a is not divisible by p, then:
   a^(p - 1) = 1 (mod p)
   Here, a = 7 and p = 13 (which is prime).
   7^(13 - 1) = 1 (mod 13)
   7^12 = 1 (mod 13)
   
   Step 3: Divide the exponent 100 by 12.
   100 = 12 * 8 + 4
   This allows us to break down 7^100 into smaller powers.
   
   Step 4: Rewrite 7^100 using the quotient and remainder.
   7^100 = (7^12)^8 * 7^4
   Substituting 7^12 = 1 (mod 13):
   7^100 = (1)^8 * 7^4 (mod 13)
   7^100 = 7^4 (mod 13)
   
   Step 5: Compute 7^4 modulo 13.
   First, find 7^2 modulo 13:
   7^2 = 49
   Divide 49 by 13:
   49 = 13 * 3 + 10
   So, 7^2 = 10 (mod 13)
   Notice that 10 is congruent to -3 modulo 13:
   7^2 = -3 (mod 13)
   
   Next, square this result to get 7^4:
   7^4 = (7^2)^2 (mod 13)
   7^4 = (-3)^2 (mod 13)
   7^4 = 9 (mod 13)
   
   Step 6: Check the result using successive powers of 7 modulo 13.
   7^1 = 7 (mod 13)
   7^2 = 49 = 10 (mod 13)
   7^3 = 10 * 7 = 70 = 5 * 13 + 5 = 5 (mod 13)
   7^4 = 5 * 7 = 35 = 2 * 13 + 9 = 9 (mod 13)
   7^5 = 9 * 7 = 63 = 4 * 13 + 11 = 11 (mod 13)
   7^6 = 11 * 7 = 77 = 5 * 13 + 12 = -1 (mod 13)
   Since 7^6 = -1 (mod 13), we can square it:
   7^12 = (-1)^2 = 1 (mod 13), which confirms the cycle length of 12.
   Since 100 = 6 * 16 + 4:
   7^100 = (7^6)^16 * 7^4 = (-1)^16 * 9 = 1 * 9 = 9 (mod 13).
   The check confirms the answer.
   
   Final answer: 9

   353 tokens in, 780 out · 3.6 s · $0.0016 · 1 message on Pro · answered by google/gemini-3.8-flash via Google AI Studio · September 28, 2026

7. GPT-6.1 Sol

   Passed: Final answer 9: right.

   1. Use Fermat’s little theorem. Since 13 is prime and 7 is not divisible by 13:
   
   7^12 ≡ 1 (mod 13)
   
   Here, “≡” means the numbers have the same remainder when divided by 13.
   
   2. Reduce the exponent using this cycle:
   
   100 = 12 × 8 + 4
   
   7^100 = (7^12)^8 × 7^4
   
   7^100 ≡ 1^8 × 7^4 ≡ 7^4 (mod 13)
   
   3. Find the remainder of 7^4:
   
   7^4 = 2401
   
   2401 = 13 × 184 + 9
   
   Therefore, the remainder is 9.
   
   4. Check the cycle directly:
   
   7^6 = 117649 = 13 × 9050 − 1
   
   7^6 ≡ −1 (mod 13)
   
   7^100 = (7^6)^16 × 7^4 ≡ (−1)^16 × 9 ≡ 9 (mod 13)
   
   This confirms the result.
   
   Final answer: 9

   341 tokens in, 440 out (198 of them reasoning) · 5.7 s · $0.0025 · 1 message on Pro · answered by openai/gpt-6.1-sol via OpenAI · September 29, 2026

8. GPT-6 Sol

   Passed: Final answer 9: right.

   1. Find a useful power of 7 modulo 13.
   7^2 = 49
   49 ≡ 10 (mod 13)
   
   2. Continue to 7^6.
   7^3 ≡ 10 × 7 ≡ 5 (mod 13)
   7^6 ≡ 5^2 ≡ 12 ≡ -1 (mod 13)
   
   3. Break up the exponent and find 7^4.
   100 = 6 × 16 + 4
   7^4 ≡ 10^2 ≡ 9 (mod 13)
   
   4. Combine the results.
   7^100 = (7^6)^16 × 7^4
   7^100 ≡ (-1)^16 × 9 ≡ 9 (mod 13)
   
   5. Check using a different grouping.
   7^12 ≡ (-1)^2 ≡ 1 (mod 13)
   100 = 12 × 8 + 4
   7^100 ≡ 1^8 × 9 ≡ 9 (mod 13)
   
   Final answer: 9

   341 tokens in, 494 out (260 of them reasoning) · 5.0 s · $0.0028 · 1 message on Pro · answered by openai/gpt-6-sol via OpenAI · September 28, 2026

9. DeepSeek V4 Pro

   Passed: Final answer 9: right.

   1. We need the remainder when 7^100 is divided by 13, which is 7^100 mod 13.
   
   2. Since 13 is prime and 7 is not divisible by 13, Fermat's little theorem gives 7^12 ≡ 1 mod 13.
   
   3. Divide the exponent 100 by 12: 100 = 12*8 + 4.
   
   4. So 7^100 = 7^(12*8 + 4) = (7^12)^8 * 7^4.
   
   5. Using 7^12 ≡ 1 mod 13, this becomes (1)^8 * 7^4 ≡ 7^4 mod 13.
   
   6. Now compute 7^4 mod 13: 7^2 = 49 ≡ 10 mod 13, so 7^4 = (7^2)^2 ≡ 10^2 = 100 ≡ 9 mod 13.
   
   7. Check: 7^4 = 2401, and 2401 - 13*184 = 2401 - 2392 = 9, so the remainder is 9.
   
   Final answer: 9

   341 tokens in, 938 out (690 of them reasoning) · 5.1 s · $0.0038 · 1 message on Pro · answered by deepseek/deepseek-v4-pro-0813 via Wafer · October 8, 2026

10. Claude Haiku 4.5

    Passed: Final answer 9: right.

    I need to find the remainder when 7^100 is divided by 13. I'll use modular arithmetic and Fermat's Little Theorem.
    
    Step 1: Apply Fermat's Little Theorem.
    Since 13 is prime and gcd(7, 13) = 1, Fermat's Little Theorem tells us that 7^(13-1) ≡ 1 (mod 13), which means 7^12 ≡ 1 (mod 13).
    
    Step 2: Express the exponent in terms of 12.
    I need to write 100 = 12q + r where 0 ≤ r < 12.
    100 ÷ 12 = 8 remainder 4
    So 100 = 12(8) + 4, which means q = 8 and r = 4.
    
    Step 3: Rewrite 7^100 using this decomposition.
    7^100 = 7^(12·8 + 4) = (7^12)^8 · 7^4
    
    Step 4: Apply Fermat's Little Theorem to simplify.
    (7^12)^8 · 7^4 ≡ 1^8 · 7^4 ≡ 7^4 (mod 13)
    
    Step 5: Calculate 7^4 mod 13.
    7^1 = 7
    7^2 = 49 = 3(13) + 10 ≡ 10 (mod 13)
    7^4 = (7^2)^2 ≡ 10^2 = 100 (mod 13)
    
    Step 6: Reduce 100 mod 13.
    100 = 7(13) + 9 = 91 + 9
    So 100 ≡ 9 (mod 13)
    
    Step 7: Check the result by computing 7^100 mod 13 another way.
    Let me verify by computing successive powers:
    7^1 ≡ 7 (mod 13)
    7^2 ≡ 10 (mod 13)
    7^3 ≡ 7 · 10 = 70 ≡ 5 (mod 13) [since 70 = 5(13) + 5]
    7^4 ≡ 7 · 5 = 35 ≡ 9 (mod 13) [since 35 = 2(13) + 9] ✓
    
    This confirms 7^4 ≡ 9 (mod 13).
    
    Step 8: Verify using the original calculation.
    Since 7^12 ≡ 1 (mod 13) and 100 = 96 + 4 = 12(8) + 4, we have 7^100 ≡ 7^4 ≡ 9 (mod 13). ✓
    
    Final answer: 9

    359 tokens in, 726 out · 4.9 s · $0.0040 · 1 message on Pro · answered by anthropic/claude-haiku-4.5 via Anthropic · September 28, 2026

11. Mistral Large 4

    Passed: Final answer 9: right.

    Step 1: Use Fermat's little theorem.
    13 is prime and gcd(7,13)=1, so 7^12 ≡ 1 (mod 13).
    
    Step 2: Reduce the exponent modulo 12.
    100 = 12*8 + 4
    so
    7^100 = 7^(12*8+4) = (7^12)^8 * 7^4.
    
    Step 3: Apply the congruence from Step 1.
    (7^12)^8 * 7^4 ≡ 1^8 * 7^4 ≡ 7^4 (mod 13).
    
    Step 4: Compute 7^4 mod 13.
    7^2 = 49
    49 ≡ 10 (mod 13)
    because 49 - 13*3 = 10.
    Therefore
    7^4 = (7^2)^2 ≡ 10^2 = 100 (mod 13).
    
    Step 5: Reduce 100 mod 13.
    100 - 13*7 = 9
    so 100 ≡ 9 (mod 13).
    
    Step 6: Combine the steps.
    7^100 ≡ 7^4 ≡ 9 (mod 13).
    
    Check: 7^4 = 2401, and 2401 = 13*184 + 9, so 7^4 leaves remainder 9. This confirms the result.
    
    Final answer: 9

    345 tokens in, 2,143 out (1,446 of them reasoning) · 33.4 s · $0.0047 · 1 message on Pro · answered by mistralai/mistral-large-4-0 via Mistral · October 9, 2026

12. Claude Sonnet 5.5

    Passed: Final answer 9: right.

    We want 7^100 mod 13. I'll use Fermat's little theorem and then check by another route.
    
    Step 1: Since 13 is prime and 7 is not a multiple of 13, Fermat's little theorem gives
    7^12 ≡ 1 (mod 13).
    
    Step 2: Divide the exponent by 12.
    100 = 12 * 8 + 4
    
    Step 3: Rewrite the power.
    7^100 = (7^12)^8 * 7^4
    ≡ 1^8 * 7^4 (mod 13)
    ≡ 7^4 (mod 13)
    
    Step 4: Compute 7^4 mod 13 in small pieces.
    7^2 = 49 = 3*13 + 10, so 7^2 ≡ 10
    7^4 ≡ 10^2 = 100 = 7*13 + 9, so 7^4 ≡ 9
    
    Step 5: Check by a different route, using repeated squaring directly on 100.
    7^2 ≡ 10
    7^4 ≡ 9
    7^8 ≡ 81 = 6*13 + 3, so 7^8 ≡ 3
    7^16 ≡ 9
    7^32 ≡ 81 ≡ 3
    7^64 ≡ 9
    Now 100 = 64 + 32 + 4, so
    7^100 ≡ 7^64 * 7^32 * 7^4 ≡ 9 * 3 * 9 = 243
    243 = 18*13 + 9, so 243 ≡ 9.
    
    Both methods agree.
    
    Final answer: 9

    447 tokens in, 512 out · 3.5 s · $0.0060 · 1 message on Pro · answered by anthropic/claude-sonnet-5.5 via Anthropic · September 28, 2026

13. Claude Sonnet 5

    Passed: Final answer 9: right.

    1. We need 7^100 mod 13. By Fermat's Little Theorem, since 13 is prime and gcd(7,13)=1, 7^12 ≡ 1 (mod 13).
    
    2. Write 100 = 12·8 + 4, so 7^100 = (7^12)^8 · 7^4 ≡ 1^8 · 7^4 ≡ 7^4 (mod 13).
    
    3. Compute 7^4: 7^2 = 49, and 49 mod 13 = 49 - 39 = 10, so 7^2 ≡ 10 (mod 13).
    
    4. Then 7^4 = (7^2)^2 ≡ 10^2 = 100 (mod 13).
    
    5. Reduce 100 mod 13: 13·7 = 91, so 100 - 91 = 9. Thus 7^4 ≡ 9 (mod 13).
    
    6. Check: compute directly 7^4 = 2401. Dividing, 13·184 = 2392, remainder 2401 - 2392 = 9. Matches step 5.
    
    Final answer: 9

    445 tokens in, 535 out (178 of them reasoning) · 6.9 s · $0.0062 · 1 message on Pro · answered by anthropic/claude-sonnet-5 via Anthropic · September 28, 2026

14. Grok 4.7

    Passed: Final answer 9: right.

    1. We need the remainder of 7^100 divided by 13, which is 7^100 mod 13.
    2. 13 is prime and does not divide 7, so Fermat's Little Theorem gives
    7^12 ≡ 1 mod 13.
    3. Divide the exponent 100 by 12:
    100 = 12*8 + 4.
    4. Therefore
    7^100 = (7^12)^8 * 7^4 ≡ 1^8 * 7^4 ≡ 7^4 mod 13.
    5. Compute the power step by step:
    7^2 = 49
    49 - 3*13 = 49 - 39 = 10,
    so 7^2 ≡ 10 mod 13.
    6. Then
    7^4 = (7^2)^2 ≡ 10^2 = 100 mod 13
    100 - 7*13 = 100 - 91 = 9,
    so 7^4 ≡ 9 mod 13.
    7. Check by another route: successive powers give
    7^1 ≡ 7,
    7^2 ≡ 10,
    7^3 ≡ 5,
    7^4 ≡ 9 mod 13,
    and also 7^6 ≡ -1, so 7^12 ≡ 1, confirming the reduction and the value 9.
    Final answer: 9

    1,579 tokens in, 803 out (515 of them reasoning) · 9.9 s · $0.0062 · 1 message on Pro · answered by x-ai/grok-4.7 via xAI · October 2, 2026

15. Kimi K3

    Passed: Final answer 9: right.

    1. 13 is prime and gcd(7, 13) = 1, so by Fermat's little theorem, 7^12 ≡ 1 (mod 13).
    
    2. Write 100 as a multiple of 12 plus a remainder:
    100 = 12·8 + 4.
    
    3. Use that to reduce the exponent:
    7^100 = 7^(12·8 + 4) = (7^12)^8 · 7^4.
    Since 7^12 ≡ 1 (mod 13), this is congruent to 1^8 · 7^4 ≡ 7^4 (mod 13).
    
    4. Compute 7^4 (mod 13) in small pieces:
    7^2 = 49 ≡ 49 − 39 = 10 (mod 13).
    7^4 = (7^2)^2 ≡ 10^2 = 100 ≡ 100 − 91 = 9 (mod 13).
    
    5. So 7^100 ≡ 9 (mod 13), meaning the remainder is 9.
    
    Check: powers cycle with period 12 since 7^12 ≡ 1, and 100 mod 12 = 4. The start of the cycle is 7^1 ≡ 7, 7^2 ≡ 10, 7^3 ≡ 5, 7^4 ≡ 9 (mod 13), which matches step 4.
    
    Final answer: 9

    428 tokens in, 655 out (339 of them reasoning) · 20.3 s · $0.0063 · 1 message on Pro · answered by moonshotai/kimi-k3 via Wafer · September 28, 2026

16. GPT-6 Astra

    Passed: Final answer 9: right.

    1. Find a useful small power of 7 modulo 13.
    
    7^2 = 49 = 13 × 3 + 10
    
    7^3 ≡ 7 × 10 = 70 ≡ 5 (mod 13)
    
    Here, ≡ means the numbers have the same remainder when divided by 13.
    
    2. Square to find the remainder of 7^6.
    
    7^6 ≡ 5^2 = 25 ≡ -1 (mod 13)
    
    Squaring again gives:
    
    7^12 ≡ (-1)^2 = 1 (mod 13)
    
    3. Split the exponent into a multiple of 12 and a leftover.
    
    100 = 12 × 8 + 4
    
    7^100 = (7^12)^8 × 7^4
    
    7^100 ≡ 1^8 × 7^4 (mod 13)
    
    4. Calculate the remaining power.
    
    7^4 ≡ (7^2)^2 ≡ 10^2 = 100 (mod 13)
    
    100 = 13 × 7 + 9
    
    So the remainder is 9.
    
    5. Check using a different exponent split.
    
    100 = 6 × 16 + 4
    
    7^100 ≡ (7^6)^16 × 7^4 ≡ (-1)^16 × 9 = 9 (mod 13)
    
    This confirms the result, and 9 is between 0 and 12 as required.
    
    Final answer: 9

    341 tokens in, 316 out · 3.9 s · $0.0096 · 1 message on Pro · answered by openai/gpt-6-astra via OpenAI · September 28, 2026

17. Gemini 3.1 Pro

    Passed: Final answer 9: right.

    To find the remainder when 7^100 is divided by 13, we can use a rule in number theory called Fermat's Little Theorem. 
    
    1. Fermat's Little Theorem states that for any prime number p and an integer a that is not divisible by p, a^(p-1) leaves a remainder of 1 when divided by p.
    In modular arithmetic notation, a^(p-1) = 1 (mod p).
    
    2. Apply the theorem with our specific numbers. Our divisor, 13, is a prime number, and our base, 7, is not a multiple of 13.
    p = 13
    a = 7
    7^(13-1) = 1 (mod 13)
    7^12 = 1 (mod 13)
    This tells us that every time we multiply 7 by itself 12 times, the remainder when divided by 13 is 1.
    
    3. Figure out how many groups of 12 are in our exponent, 100, by dividing 100 by 12.
    100 / 12 = 8 with a remainder of 4
    This means we can write 100 as (12 * 8) + 4.
    
    4. Rewrite 7^100 using the rules of exponents to separate the groups of 12.
    7^100 = 7^((12 * 8) + 4)
    7^100 = (7^12)^8 * 7^4
    
    5. Substitute the remainder of 1 for 7^12 to simplify the problem.
    (7^12)^8 * 7^4 = (1)^8 * 7^4 (mod 13)
    1 * 7^4 = 7^4 (mod 13)
    So, finding the remainder of 7^100 divided by 13 is exactly the same as finding the remainder of 7^4 divided by 13.
    
    6. Calculate the remainder of 7^4 divided by 13 step by step. First, find 7^2.
    7^2 = 49
    Divide 49 by 13 to find the remainder. 
    13 * 3 = 39
    49 - 39 = 10
    So, 7^2 leaves a remainder of 10.
    
    7. Use the remainder of 7^2 to easily find the remainder of 7^4. 
    7^4 = (7^2)^2
    Using our remainder from the previous step:
    (7^2)^2 = 10^2 (mod 13)
    10^2 = 100
    
    8. Divide 100 by 13 to find our final remainder.
    13 * 7 = 91
    100 - 91 = 9
    Therefore, 7^4 (and thus 7^100) leaves a remainder of 9 when divided by 13.
    
    9. Let's check our result by calculating the powers of 7 one by one instead of squaring the remainder.
    7^1 = 7 (mod 13)
    7^2 = 49, and 49 = 39 + 10, so it's 10 (mod 13)
    7^3 = 7^2 * 7 = 10 * 7 = 70. Divide 70 by 13: 13 * 5 = 65, and 70 - 65 = 5. So 7^3 is 5 (mod 13).
    7^4 = 7^3 * 7 = 5 * 7 = 35. Divide 35 by 13: 13 * 2 = 26, and 35 - 26 = 9. So 7^4 is 9 (mod 13).
    The calculations match perfectly, confirming our remainder is correct.
    
    Final answer: 9

    353 tokens in, 2,525 out (1,620 of them reasoning) · 21.3 s · $0.0155 · 1 message on Pro · answered by google/gemini-3.1-pro-preview via Google AI Studio · September 28, 2026

18. Claude Opus 5.5

    Passed: Final answer 9: right.

    1. Use Fermat's little theorem.
    13 is prime and 13 does not divide 7, so:
    7^12 ≡ 1 (mod 13)
    
    2. Write the exponent 100 in terms of multiples of 12.
    100 = 12 · 8 + 4
    
    3. Rewrite 7^100 using this split.
    7^100 = (7^12)^8 · 7^4
    Since 7^12 ≡ 1, this gives:
    7^100 ≡ 1^8 · 7^4 (mod 13)
    7^100 ≡ 7^4 (mod 13)
    
    4. Compute 7^4 mod 13 by repeated squaring.
    First square 7:
    7^2 = 49
    49 − 3·13 = 49 − 39 = 10
    So 7^2 ≡ 10 (mod 13).
    Then square again:
    7^4 ≡ 10^2 = 100
    100 − 7·13 = 100 − 91 = 9
    So 7^4 ≡ 9 (mod 13).
    
    5. Conclude.
    7^100 ≡ 9 (mod 13)
    
    Check (a second route):
    First compute 7^3:
    7^3 = 7^2 · 7 ≡ 10 · 7 = 70
    70 − 65 = 5, so 7^3 ≡ 5 (mod 13).
    Then square it:
    7^6 = (7^3)^2 ≡ 5^2 = 25
    25 − 13 = 12, so 7^6 ≡ 12 ≡ −1 (mod 13).
    Now split the exponent by 6:
    100 = 6 · 16 + 4
    7^100 = (7^6)^16 · 7^4 ≡ (−1)^16 · 9 = 9 (mod 13)
    Both routes give 9.
    
    Final answer: 9

    447 tokens in, 805 out (214 of them reasoning) · 7.2 s · $0.0179 · 1 message on Pro · answered by anthropic/claude-opus-5.5 via Google · September 28, 2026

19. Claude Fable 5.1

    Passed: Final answer 9: right.

    We want 7^100 mod 13.
    
    Step 1: Use Fermat's Little Theorem. Since 13 is prime and 13 does not divide 7,
    7^12 ≡ 1 (mod 13).
    So powers of 7 repeat with a cycle that divides 12.
    
    Step 2: Reduce the exponent modulo 12.
    100 = 12 × 8 + 4
    So 7^100 = (7^12)^8 × 7^4 ≡ 1^8 × 7^4 = 7^4 (mod 13).
    
    Step 3: Compute 7^4 mod 13 in small pieces.
    7^2 = 49
    49 = 3 × 13 + 10, so 7^2 ≡ 10 (mod 13)
    7^4 = (7^2)^2 ≡ 10^2 = 100 (mod 13)
    100 = 7 × 13 + 9, so 7^4 ≡ 9 (mod 13)
    
    Step 4: Conclude.
    7^100 ≡ 9 (mod 13)
    
    Check (using the actual order of 7 mod 13): 7^2 ≡ 10, 7^3 ≡ 70 ≡ 5, 7^4 ≡ 35 ≡ 9, 7^5 ≡ 63 ≡ 11, 7^6 ≡ 77 ≡ 12 ≡ -1, so 7^12 ≡ 1, confirming the cycle length divides 12. Then 7^96 = (7^12)^8 ≡ 1 and 7^100 ≡ 7^4 ≡ 9. Consistent.
    
    Final answer: 9

    447 tokens in, 536 out · 6.4 s · $0.0313 · 1 message on Pro · answered by anthropic/claude-fable-5.1 via Google · September 28, 2026

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