Question The remainder of 5^22+6 when divided by 8 is:

Solution

There are many approaches to this question a few are straight forward other more involved.

The idea is to write 5^22 in a format such that most of the terms are multiple of 8.

So we get 5^22 can written as (1+4)^22=1+{::}^22C_1xx4+{::}^22C_2xx4^2+...+{::}^22C_22xx4^22

So the remained will be 1.

While this needs some foresight, others thought 5^22=25^11=(24+1)^22. Now since one term is a multiple of 8, on dividing it with 8 we will get a remainder of 1.

Now we come to other methods:

Observe that the last 3 digits of 5^22 are 625, so on dividing by 8 we get a remainder of 1.

Alternatively you solve this by modulo which is 5^22mod8 =((25mod8)xx..xx(25mod8))mod8 =1mod8