5¹⁷ mod 23

First step: Converting the power to binary. Dividing by 2 we get the following.

^{We get the following binary:: (10001)}₂

Step 2: Using the repeated square algorithm.

5^{2^0}   mod 23 à 5

Squaring the result we get,

1. 5²  mod 23 à 25 mod 23 à 2

Again, squaring the result

2² mod 23 à 4 mod 23 à 4

Squaring again,

4² mod 23 à 16 mod 23 à 16

Squaring result again,

16² mod 23 à 256 mod 23 à 3

Now, raising to its successive power.

 (3¹*16⁰*4⁰*2⁰*5¹) mod 23

 (3*1*1*1*5) mod 2

 15 mod 23 = 15

 In this way, repeated sqauring algorithm can be used to calculate the mod of a higher number without using a calculator.