What is the remainder when 132^5 2*132^4 + 6*132^3 3*132 is divided

Question

What is the remainder when  132^5 − 2*132^4 + 6*132^3 −3*132  is divided by 65?

A. 40
B. 42
C. 44
D. 47
E. 54

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gmatophobia · Accepted Answer

Remainder when  \frac{132^5 }{ 65}  =  2^5

Remainder when  \frac{132^4 }{ 65}  =  2^4

Remainder when  \frac{132^3 }{ 65}  =  2^3

Remainder when  \frac{132 }{ 65}  = 2

Net remainder =

2^5 - 2* 2^4 + 6*2^3 - 3 * 2 = 6 (2^3 - 1) = 7 * 6 = 42

Option B

sumit99kr · Answer

The above expression can be re-written as below-

(130+2)^5 - 2*(130+2)^4 + 6*(130+2)^3 - 3(130+2)

Remainder of each term when divide by 65 will be as below:

2^5 - 2*(2^4) + 6*(2^3) - 3(2) 
32 - 32 + 48 - 6 = 42.
So, the remainder is 42.

Hence B is the correct choice

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BrushMyQuant · Answer

We need to find the remainder of  132^5 − 2*132^4 + 6*132^3 −3*132  by 65

We solve these problems by using Binomial Theorem, where we split the number into two parts, one part is a multiple of the divisor(65 or 130) and a big number, other part is a small number.

=>  132^{5}  =  (130 + 2)^{5}

Watch this video to  MASTER BINOMIAL Theorem

Now, when we expand this expression then all the terms except the last term will be a multiple of 130 or 65. 
=> All terms except the last term will give 0 as remainder then divided by 65
=> Problem is reduced to what is the remainder when the last term is divided by 65
=> What is the remainder when  5C5 * 130^0 * 2^{5}  is divided by 65 = Remainder of  2^{5}  by 65 = 32

=> Remainder of  132^5   by 65 = remainder of 32 by 65 = 32

Remainder of 2*132^4 by 65 =  Remainder of 2*2^4 by 65 = Remainder of 32 by 65 = 32

Remainder of 6*132^3 by 65 =  Remainder of 6*2^3 by 65 = Remainder of 48 by 65 = 48

Remainder of 3*132 by 65 =  Remainder of 3*(130 + 2) by 65 = Remainder of 3*2 by 65 = 6

=> Total Remainder = 32 - 32 + 48 - 6 = 42

So,  Answer will be B 
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

bumpbot · Answer

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What is the remainder when 132^5 2132^4 + 6132^3 3*132 is divided

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