Binomial, divisibility and other approaches

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


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

Let us start with the most common approach:

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!!

So the answer is 1+6=7

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