## When (67^67 + 67) is divided by 68 then what is the remainder?

Ronnie
Posts: 2
Joined: Sun Jan 24, 2016 3:27 pm

### When (67^67 + 67) is divided by 68 then what is the remainder?

When (6767 + 67) is divided by 68 then what is the remainder?
Last edited by Ronnie on Sun Jan 24, 2016 5:13 pm, edited 1 time in total.
PrepareBetter.in
Posts: 2
Joined: Thu Jan 21, 2016 2:17 pm

### Re: When (67^67 + 67) is divided by 68 then what is the remainder?

There are mainly two methods to solve this problem.
1) By polynomial method

Let us see the polynomial expansions...

an + bn = (a + b)(b0an-1 - b1an-2 + ... - bn-2a1 + bn-1a0) (For n>1 and odd)

an - bn = (a - b)(b0an-1 + b1an-2 + ... + bn-2a1 + bn-1a0)

So from above polynomial expansions we can say that
(xn + 1) is divisible by (x + 1) only when n is odd

So (6767 + 1) is divisible by (67 + 1)
i.e (6767 + 1) is divisible by 68

Now for the given problem we can write
(6767 + 67) = (6767 + 1) + 66

but (6767 + 1) is divisible by 68

So when (6767 + 1) + 66 is divided by 68 then it will give 66 as remainder.

2)By trial and error method

22 + 2 divided by 3
= 4 + 2
=6
so when 6 divided by 3 then remainder = 0

So by doing calculations like this we can get

22 + 2 divided by 3 then remainder = 0
33 + 3 divided by 4 then remainder = 2
44 + 4 divided by 5 then remainder = 0
55 + 5 divided by 6 then remainder = 4

So by observing above examples we can say

xx + x is divided by (x+1) then the remainder is (x-1) where x is odd number

So now we can say when 6767 + 67 is divided by 68 then remainder is 66.
Ronnie
Posts: 2
Joined: Sun Jan 24, 2016 3:27 pm

### Re: When (67^67 + 67) is divided by 68 then what is the remainder?

Thank you
Santhosh
Posts: 1
Joined: Sat Feb 20, 2016 1:05 pm

good post

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