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

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

Postby Ronnie » Sun Jan 24, 2016 4:03 pm

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

Postby PrepareBetter.in » Sun Jan 24, 2016 5:02 pm

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

Let us see the polynomial expansions...

a + b = (a + b)(ba - ba + ... - ba + ba) (For n>1 and odd)

a - b = (a - b)(ba + ba + ... + ba + ba)

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

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

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

but (67 + 1) is divisible by 68. So remainder for first term (67 + 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 (67 + 1) + 66 is divided by 68 then it will give 66 as remainder.

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

2) By using

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) = x + x
c = −1

By applying The Polynomial Remainder Theorem

f(−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.

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3) By trial and error method

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

3 + 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

2 + 2 divided by 3 then remainder = 0
3 + 3 divided by 4 then remainder = 2
4 + 4 divided by 5 then remainder = 0
5 + 5 divided by 6 then remainder = 4
6 + 6 divided by 7 then remainder = 0

So by observing above examples we can say

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

So now we can say when 67 + 67 is divided by 68 then remainder is 66.
Ronnie
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Re: When (67^67 + 67) is divided by 68 then what is the remainder?

Postby Ronnie » Sun Jan 24, 2016 5:13 pm

Thank you :D
Santhosh
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Re: When (67^67 + 67) is divided by 68 then what is the remainder?

Postby Santhosh » Sat Feb 20, 2016 3:51 pm

good post

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