When 7^2025 is divided by 13, the remainder is.
Answer and solution
Correct answer: 8
Solution
Answer: 8
1. The powers of 7 repeat in a cycle when divided by 13.
2. 7¹ leaves remainder 7.
7² leaves remainder 49, which is 10.
7³ leaves remainder 5.
7⁴ leaves remainder 9.
Continuing this pattern, 7¹² leaves remainder 1. So the cycle repeats after every 12 powers.
3. Now divide 2025 by 12:
4. 2025 = 12 × 168 + 9
5. Therefore, 7²⁰²⁵ leaves the same remainder as 7⁹.
6. Since 7³ leaves remainder 5 and 7⁶ leaves remainder 12,
7. 7⁹ has the same remainder as 12 × 5 = 60.
8. 60 divided by 13 leaves remainder 8.
9. Answer: 8 (TITA).
Why this method works:
Modular arithmetic keeps only remainders. From 7³≡5, squaring gives 7⁶≡25≡−1 modulo 13. Squaring again gives 7¹²≡1, providing the cycle without calculating a huge power.
Independent check and common trap:
Since 2025=6×337+3 and 337 is odd, the remainder is (−1)³³⁷×5≡−5≡8. Always convert a negative residue into the standard remainder from 0 to 12.
Apply: 7⁶≡−1 mod 13 and 2025=6×337+3. Therefore 7²⁰²⁵≡−7³≡−5≡8.
Check: Since 2025=6×337+3 and 337 is odd, the remainder is (−1)³³⁷×5≡−5≡8. Always convert a negative residue into the standard remainder from 0 to 12.
1. The powers of 7 repeat in a cycle when divided by 13.
2. 7¹ leaves remainder 7.
7² leaves remainder 49, which is 10.
7³ leaves remainder 5.
7⁴ leaves remainder 9.
Continuing this pattern, 7¹² leaves remainder 1. So the cycle repeats after every 12 powers.
3. Now divide 2025 by 12:
4. 2025 = 12 × 168 + 9
5. Therefore, 7²⁰²⁵ leaves the same remainder as 7⁹.
6. Since 7³ leaves remainder 5 and 7⁶ leaves remainder 12,
7. 7⁹ has the same remainder as 12 × 5 = 60.
8. 60 divided by 13 leaves remainder 8.
9. Answer: 8 (TITA).
Why this method works:
Modular arithmetic keeps only remainders. From 7³≡5, squaring gives 7⁶≡25≡−1 modulo 13. Squaring again gives 7¹²≡1, providing the cycle without calculating a huge power.
Independent check and common trap:
Since 2025=6×337+3 and 337 is odd, the remainder is (−1)³³⁷×5≡−5≡8. Always convert a negative residue into the standard remainder from 0 to 12.
G Strategy shortcut
Spot: Number System – Remainders.Apply: 7⁶≡−1 mod 13 and 2025=6×337+3. Therefore 7²⁰²⁵≡−7³≡−5≡8.
Check: Since 2025=6×337+3 and 337 is odd, the remainder is (−1)³³⁷×5≡−5≡8. Always convert a negative residue into the standard remainder from 0 to 12.
G Strategy: Spot: Number System – Remainders. Apply: 7⁶≡−1 mod 13 and 2025=6×337+3. Therefore 7²⁰²⁵≡−7³≡−5≡8. Check: Since 2025=6×337+3 and 337 is odd, the remainder is (−1)³³⁷×5≡−5≡8. Always convert a negative residue into the standard remainder from 0 to 12.