What is the remainder when (67^67 + 67) is divided by 68 ?

Question

(A) 1 
(B) 11
(C) 63
(D) 66 
(E) 67

rajatchopra1994 · Answer

Explanation:

(67^67)/ 68 + 67/68
(-1)^67/68 + 67/68
-1 + 67 / 68
66/68

Remainder = 66

IMO-D

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1473 · Answer

because 67 and 68 are coprimes, we should not look to simplify 67 raised to any power divided by 68 but instead look directly at the sum. (67^n +67)/68 gives you 66 as remainder.

hence, in my opinion the answer is D - 66.

AliceSheetal · Answer

=67(67+1) =67*68
divisor=68
(67*68)/68
=R-67

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=EQdpYr3SACM

We need to find the remainder when  67^{67} + 67  is divided by 68?

Remainder of  67^ {67} + 67  by 68 = Remainder of    67^ {67}   by 68 + Remainder of 67 by 68

To find Remainder of  67^ {67}   by 68 we will be using Binomial Theorem, where we split the number into two parts, one part is a multiple of the divisor(68) and a big number, other part is a small number.

=>  67^{67}  =  (68 - 1)^{67}

Watch this video to   MASTER BINOMIAL Theorem

Now, if we use Binomial Theorem to expand this then all the terms except the last term will be a multiple of 68 
 => All terms except the last term will give remainder of 0 when divided by 68 
=> Problem is reduced to what is the remainder when the last term (i.e.  67C67 * 68^0 * (-1)^{67} ) is divided by 68
=> Remainder of -1 is divided by 68 = 67

Remainder of  67^{67} + 67  by 68   =  Remainder of  67^{67}  by 68 + Remainder of 67 by 68 = Remainder of 67 + 67 = Remainder of 68 + 66
= 0 + 66 = 66

So,  Answer will be D 
Hope it Helps!

Watch following video to MASTER Remainders by 2, 3, 5, 9, 10 and Binomial Theorem

https://www.youtube.com/watch?v=h1HTEhmUb1w

anurag029 · Answer

Hi  rajatchopra1994 , can you explain the transition between the first and the second step?

BrushMyQuant · Answer

anurag029 
That happened because of binomial theorem which I explained in my post. Once you are familiar with the concept then you can directly write remainder of 67^power by 68 as Remainder of -1^power by 68.
-1 comes by doing 67-68

Hope it helps!

What is the remainder when (67^67 + 67) is divided by 68 ?

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What is the remainder when (67^67 + 67) is divided by 68 ?

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