help number theory

5^(301)  mod 7==5 - the remainder

Hello Guest,

there is a calculator, namely: Remainder Calculator | The best Long Division Calculator (calculatored.com)

and the link https://www.calculatored.com/math/algebra/remainder-calculator.

I also got the remainder 5.

Hope that was helpful.^^

Straight

What is the remainder when 5^(301) is divided by 7?

Let's look for a pattern

5^1              5     mod7 = 5                5^(6n+1) = 5 mod 7

5^2              25     mod 7 = 4            5^(6n+2) = 4  mod 7

5^3            125   mod 7 = 6             5^(6n+3) = 6  mod 7

5^4            625  mod 7 = 2              5^(6n+4) = 2  mod 7

5^5            3125 mod 7 =3              5^(6n+5) = 3  mod 7

5^6           15625  mod 7 = 1           5^(6n) = 1  mod 7

7         78125 mod 7 = 5

8         390625 mod 7 =4

an so we have a pattern

301 = 1 mod 6  

so the remainder will be 5