Re: What is the remainder when 43^86 is divided by 5?

An easy approach that will help you solve the question in less than a minute.

Now, we have 43^86 divided by 5

5*9= 45

If i divide, 43 by 5 i get the remainder as 3 or -2.
I hope you get the -2 logic. ( 3-5 => -2)

remainder can be 3 or it can also be -2.


Now,using this approach. 43^86 can be written as (-2)^86

=> 43^86
=> (-2)^86
=> (-2)^84 * (-2)^2 ------ When the bases are same the powers get added, we can split the term -2^86
=> (-2^4)^21 * (-2)^2 ----------- 84 can be rewritten as 4*21 and the 4 is brought inside the braces making -2 as -2^4 ---- remember this is all about playing with powers... so do it carefully....

=> 16^21 * 4

16^21 * 4 divided by 5
16 when divided by 5 gives remainder 1

1^21 *4
=> 4 is the remainder...

You must be wondering when 86 was splitted to 84 and 2. See, we have to figure the power of 2 in such a way that it is easily divisible by 5.
2^1=2, when divided by 5 will give remainder 2...not of our use
2^2=4, when divided by 5 will give remainder 4...not of our use... this can be used to deduce the remained-- it is one of my favourite's apporach, but i will explain that later...
2^3=8, when divided by 5 will give remainder 3...not of our use
2^4=16, when divided by 5 will give remainder 1... this one is good.

Therefore, we get the remainder as 4..

Just practice few questions using this approach and it wont take more than a minute to solve remainder types of questions....
You can use the same approach for any number.