\(A=3+3^2+3^3+3...+3^{31}\\ =\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{28}+3^{29}+3^{30}\right)+3^{31}\\ =\left(3+3^2+3^3\right)+3^3\left(3+3^2+3^3\right)+...+3^{27}\left(3+3^2+3^3\right)+3^{31}\\ =39+3^3.39+...+3^{27}.39+3^{31}\\ =39.\left(1+3^3+...+3^{27}\right)+3^{31}\\ Mà:39.\left(1+3^3+...+3^{27}\right)⋮13\left(Do:39⋮13\right)\\ Mà:3^{31}:13\left(dư:3\right)\\ Vậy:39.\left(1+3^3+...+3^{27}\right)+3^{31}:13\left(dư:3\right)\\ \Rightarrow A:13\left(dư:3\right)\)
A = 3 + 3² + 3³ + ... + 3³¹
= 3 + 3² + 3³ + 3⁴ + 3⁵ + 3⁶ + 3⁷ + ... + 3²⁹ + 3³⁰ + 3³¹
= 3 + (3² + 3³ + 3⁴) + (3⁵ + 3⁶ + 3⁷) + ... + (3²⁹ + 3³⁰ + 3³¹)
= 3 + 3².(1 + 3 + 3²) + 3⁵.(1 + 3 + 3²) + ... + 3²⁹.(1 + 3 + 3²)
= 3 + 3².13 + 3⁵.13 + ... + 3²⁹.13
= 3 + 13.(3² + 3⁵ + ... + 3³¹)
Do 13.(3² + 3⁵ + ... + 3³¹) ⋮ 13
⇒ 3 + 13.(3² + 3⁵ + ... + 3³¹) chia 13 dư 3
Vậy A chia 13 dư 3
