Numerical answer
What is the remainder when 7¹⁰⁰ is divided by 13?
Show answer and solution
Answer: 9
By Fermat's little theorem, 7¹² ≡ 1 (mod 13).
100 = 12 × 8 + 4, so 7¹⁰⁰ ≡ 7⁴ (mod 13).
7² = 49 ≡ 49 − 39 = 10 ≡ −3. Then 7⁴ ≡ (−3)² = 9 (mod 13).
Remainder = 9.
Tip: Using negative remainders (49 ≡ −3) keeps the squaring small.
100 = 12 × 8 + 4, so 7¹⁰⁰ ≡ 7⁴ (mod 13).
7² = 49 ≡ 49 − 39 = 10 ≡ −3. Then 7⁴ ≡ (−3)² = 9 (mod 13).
Remainder = 9.
Tip: Using negative remainders (49 ≡ −3) keeps the squaring small.
Saved on this device. Sign in or create a free account to keep it on every device.