Question

[Ngô Hồng Thuận]

Tìm số dư khi chia \(S=2^5+2^{10}+2^{15}+...+2^{45}+2^{50}\) cho 30.

Answer 1

[Mr Lazy]

Ta có: \(2^5+2^{10}+2^{15}+2^{20}=1082400=30.36080\)

\(S=\left(2^5+2^{10}+2^{15}+2^{20}\right)+\left(2^{25}+2^{30}+2^{35}+2^{40}\right)+2^{45}+2^{50}\)

\(=\left(2^5+2^{10}+2^{15}+2^{20}\right)\left(1+2^{20}\right)+2^{45}+2^{30}\)

\(=30.36080\left(2^{20}+1\right)+2^{45}+2^{50}\)

Xét 2⁴⁵ và 2⁵⁰

+2²⁰ ≡ 16 (mod 30)
+2²⁵ ≡ 2 (mod 30) 
+2⁴⁵ = 2²⁰ . 2²⁵ ≡ 16.2 = 32 ≡ 2 (mod 30)
+2⁵⁰ = (2²⁵)²  ≡ 2²  ≡ 4 (mod 30)

=> 2⁴⁵ + 2⁵⁰ ≡ 2 + 4 ≡ 6 (mod 30)

=> 2⁴⁵ + 2⁵⁰ chia 30 dư 6.

Suy ra S chia 30 dư 6.