What us the remainder if 9^11 is divided by 1000 ?

Question

What is the remainder if 9^11 is divided by 1000 ?

A. 509
B. 561
C. 609
D. 969
E. 999

chetan2u · Accepted Answer

We can tackle this by expansion.

As remainder by 1000 means last 3 digits, we are concerned with only last 3 digits.

9^{11}=(10-1)^{11}

expansion of the term  (a-b)^n=nC0*a^0*(-b)^n+nC1*a^1*(-b)^{n-1}+..... 
when b is 1 and n is odd, the expansion becomes =  (a-1)^{odd}=-nC0*a^n+nC1*a^1+...

(10-1)^{11}=-11C0*10^0+11C1*10^1-11C2*10^2+..... 
We will drop remaining terms as each will be divisible by 1000
=> =-1+110-5500=-5391 or -391
We cannot have a negative remainder, so 1000-391=609

C

CrackverbalGMAT · Answer

If you are good at multiplying, then you might try this. This method is for this case as the powers are small.

R  = R  = R  * R  * R

9^4 = 6561  gives a remainder of 561, when divided by 1000

So, The above equation comes down as  R  * R  * R

561 * 561 = 314721 which gives a remainder of 721 when divided by 1000

The expression is now  R  * R

729 * 721 = 525609 which gives a remainder of 609 when divided by 1000

Option C

Arun Kumar

BrushMyQuant · Answer

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

https://www.youtube.com/watch?v=g0-jL39d2cM

We need to find What is the remainder if 9^11 is divided by 1000?

Theory : Remainder of any number by 1000 will be equal to the last three digits of the number.

1000 =  10^3

Now, if we are able to express  9^{11}  in terms of powers of 10 then any number with power of ≥ 3 will give zero remainder when divided by  10^3

9^{11}  =  (10-1)^{11}

Now,  (10-1)^{11}  can be opened using Binomial theorem

=>  (10-1)^{11}  =  11C0 * 10^{11}(-1)^0 + 11C1 * 10^{10}(-1)^1 + 11C2 * 10^9(-1)^2 + .... + 11C8 * 10^{3}(-1)^8 + 11C9 * 10^{2}(-1)^9 + 11C10 * 10^{1}(-1)^{10} + 11C11 * 10^{0}(-1)^{11}

Now, all the terms starting from the first time till the  4^{th}  last term will have a power of 10 greater than 3
=> Their remainder by  10^3  will be 0

=> Our problem is reduced to what is the remainder when  11C9 * 10^{2}(-1)^9 + 11C10 * 10^{1}(-1)^{10} + 11C11 * 10^{0}(-1)^{11}  is divided by 1000

=>  \frac{11*10}{2}*100*-1 + 11*10*1 + 1*1*-1  divided by 1000
=> -5500 + 110 - 1 divided by 1000
=> -5391 divided by 1000
=> Remainder will be same as remainder of -391 by 1000
=> 1000 - 391 = 609

So,  Answer will be C 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

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What us the remainder if 9^11 is divided by 1000 ?

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What us the remainder if 9^11 is divided by 1000 ?

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