Problem 9 · Classify with modular arithmetic · 5 points
Made by AI and not reviewed by a real person.
When an integer’s fourth power is divided by 65, which of these numbers cannot be the remainder?
- 35
- 56
- 26
- 16
- 61
Show a hint
- Try checking fourth-power remainders using the prime factors of 65 rather than calculating large powers.
- For a base not divisible by 13, Fermat’s little theorem tells you what its twelfth power is modulo 13.
Show the answer and solution
Answer: B — 56
The remainders 16, 26, 35, and 61 occur for bases 2, 13, 20, and 4, respectively. But if x⁴ ≡ 56 (mod 65), then x⁴ ≡ 4 (mod 13), so 13 does not divide x. Fermat’s little theorem would give 1 ≡ x¹² ≡ 4³ ≡ −1 (mod 13), a contradiction. The trap is that 4⁴ = 256 ends in 56, but its remainder modulo 65 is 61.