Question

Amélie

what is the remainder when 2^20+3^30+4^40+5^50 is divided by 7?

同餘

Answer 1

qn

由費馬小定理可知，對於質數p， aᵖ⁻¹≡1 (mod p)
→ a⁶≡1 (mod 7)
所以 2²⁰+3³⁰+4⁴⁰+5⁵⁰ = (2⁶)³·2²+(3⁶)⁵+(2⁶)¹³·2²+(5⁶)⁸5²
≡ 4+1+4+25 (mod 7)
≡ 6 (mod 7)