## 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 three 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 remainder for first term (6767 + 1) is 0.

And remainder for second term 66 is equal to 66. (66/68 then remainder = 66).

So total remainder = 0 + 66 = 66.

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

Hence when (6767 + 67) is divided by 68 then remainder is 66.

2) By using The Polynomial Remainder Theorem

According to The Polynomial Remainder Theorem
"If a polynomial f(x) is divided by another polynomial (x−c) then the remainder is always equal to f(c)"

So Given problem equation can be written as

Lets assume 67 = x. Above equation can be written as

Here, to apply The Polynomial Remainder Theorem,
f(x) = xx + x
c = −1

By applying The Polynomial Remainder Theorem

f(−1) = −1−1 + (−1)
f(−1) = −2

Here the remainder we got is negative. But remainder can not be negative. So to convert negative remainder into positive, divisor is added to negative remainder. So,
−2 + 68 = 66

So the remainder is 66.

To know more click The Polynomial Reminder Theorem

3) By trial and error method

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

33 + 3 divided by 4
= 27 + 3
=30
so when 30 divided by 4 then remainder = 2

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
66 + 6 divided by 7 then remainder = 0

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