Question

[Huỳnh Thị Mỹ Linh]

Cho S= 1+5¹+5²+...+5¹²+5¹³

Hỏi S chia cho 31 dư bao nhiêu

Answer 1

[l҉o҉n҉g҉ d҉z҉]

Ta có : S = 1 + 5¹ + 5² +...+ 5¹² + 5¹³ 

=> S = 1 + 5 + (5² + 5³ + 5⁴ ) + (5⁵ + 5⁶ + 5⁷) + ...... + (5¹¹ + 5¹² + 5¹³) 

=> S = 6 + 5²(1 + 5 + 25) + 5⁵(1 + 5 + 25) + ..... + 5¹¹(1 + 5 + 25)

=> S = 6 + 5².31 + 5⁵.31 + ..... + 5¹¹.31

=> S = 6 + (5².31 + 5⁵.31 + ..... + 5¹¹.31)

=> S = 6 + 31(5² + 5⁵ + ..... + 5¹¹)

Mà : 31(5² + 5⁵ + ..... + 5¹¹) chia hết cho 31

Nên S = 6 + 31(5² + 5⁵ + ..... + 5¹¹) chia 31 dư 6 

Answer 2

[alan walker]

5S= 5+5² +5³ +....+ 5¹³ + 5¹⁴

4S=(5+5² +5³ +....+ 5¹³ + 5¹⁴) - (1+ 5 + 5²+5³ +....+ 5¹² + 5¹³ )

4S= 5¹⁴ - 1 

S=  5¹⁴ - 1 :4 =6103515625 -\(\frac{1}{4}\)= 6103515624.75

S: 31 = 6103515624.75 : 31