What is the remainder when 123^4 - 123^3 - 123^2 - 123 is divided by

Question

What is the remainder when  123^4 - 123^3 - 123^2 - 123  is divided by 122?

A. 0
B. 118
C. 119
D. 120
E. 121

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chetan2u · Accepted Answer

123^4 - 123^3 - 123^2 - 123  
 123^3(123-1)-123(123+1)=123^3*122-123*124 
 Remainder can be added, multiplied or subtracted.  
So we are looking for remainder when -123*124 is divided
Thus, -(122+1)(122+2) will give remainder of -1*2=-2. 
As remainder cannot be negative, the answer is 122-2 or 120.

D

gmatophobia · Answer

Remainder when  \frac{123^4 }{ 122}  = 1

Remainder when  \frac{123^3 }{ 122}  = 1

Remainder when  \frac{123^2 }{ 122}  = 1

Remainder when  \frac{123 }{ 122}  = 1

1 - 1 -1 -1 = -2

Net remainder = 122 - 2 = 120

Option D

meowmeowmeow22 · Answer

hi,

could you please explain this part: "Net remainder = 122 - 2 = 120"? 
I got -2 from the original expression but i'm not sure how you get 120 from that

gmatophobia · Answer

Hey  meowmeowmeow22

Remainders can never be negative. So if you end up getting negative remainder the net (actual) remainder is equal to divisor + (the negative remainder)

Lets take an example to illustrate this

Divisor = 7 
 Dividend = 11

We know when 11 when divided by 7 results in a quotient of 1 and remainder of 4.

\frac{11}{7}  can also be represented as  \frac{14 - 3 }{ 7}

14/7 leaves a remainder = 0

-3/7 leaves a remainder = -3

As the remainder is negative, the actual remainder is 7 - 3 = 4

Similarly, in the above question we obtained a (negative) remainder of -2. Hence the actual remainder is 122 - 2 = 120

Hope this helps.

BrushMyQuant · Answer

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

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

123^4 - 123^3 - 123^2 - 123   =  123^3(123 - 1) - 123( 123 + 1) 
=  123^3 * 122 - 123 * 124

Remainder of  123^3 * 122 - 123 * 124  by 122  = Remainder of  123^3 * 122  by 122 - Remainder of  123 * 124  by 122
= Remainder of 0 - Remainder of 123 by 122 * Remainder of 124 by 122 = 0 - 1*2 = 0 - 2= -2

Remainder of -2 by 122  will be same as -2 + 122 = 120

So,  Answer will be D 
Hope it helps!

Playlist on Solved Problems on Remainders  here  
 
 Remainders by 2, 3, 5, 9, 10 and Binomial Theorem

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

bumpbot · Answer

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What is the remainder when 123^4 - 123^3 - 123^2 - 123 is divided by

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What is the remainder when 123^4 - 123^3 - 123^2 - 123 is divided by

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