What is the remainder when 10^98-1 is divided by 11 ?

Question

Q. What is the remainder when (10^98 ) - 1 is divided by 11 ?

A. 0

B. 9

C. 11

D. 22

E. 33

XyLan · Accepted Answer

Woody19 , even if you have absolutely no idea how to approach this Q, Just try to reach the nearest number in the numerator that is divisible by the denominator. The nearest number is 99, which is perfectly divisible by the denominator 11. Thus, let's play around! 99 is divisible by 11 100 - 1 is divisible by 11 10^2 - 1 is divisible by 11 : 99 = 100 - 1 = 10^2 - 1 : 10-raised to even power - 1 Similarly, we can see the next big number is of the format: 9999 is divisible by 11 10000 - 1 is divisible by 11 10^4 - 1 is divisible by 11 : 9999 = 10000 - 1 = 10^4 - 1 : 10-raised to even power - 1 Similarly, we can see the next big number is of the format: 999999 is divisible by 11 1000000 - 1 is divisible by 11 10^6 - 1 is divisible by 11 : 999999 = 1000000 - 1 = 10^6 - 1 : 10-raised to even power - 1 Aha! It's a pattern. If we have EVEN number of 9s such as 99, 9999 or 999999, etc , in the numerator, the format is clearly divisible by 11. EVEN number of 9s = 10- raised to even power - 1 Let's bring the son-of-the-gun in the question: (10^98 - 1) | 98 is an even-power : 10-raised to even power - 1 The format is clearly divisible by 11. Remainder = 0. Hallelujah!

TakeAway: PS: Also can be taken care of by the remainder theorem. Do not be intimidated by that big number - 10^98 because GMAT tests your tenacity to reach the right answer efficiently , no matter what. Familiarise yourself with both the methods - remainder theorem and number-substitution - and tame that beast!

Staphyk · Answer

Hello this simple 
(10^2(49) -1)/11
(100^49-1)/11
We treat  remainders as dividends  :can be added ,subtracted and multiplied 
.: Rem-> (1^49-1)/11 —->
(1-1)/11—> 0/11
Hence remainder is zero

Or there is a rule that states that:
 (A^(Even))/(A+1) —> remainder is always 1
However,  (A^(odd))/(A+1) —> remainder is always A

Hence  Answer A

Posted from my mobile device

800GMAT2019 · Answer

HEY! What about this question lets us know to "Check for odd even pattern?"
How can we recognize this problem needs us to "plug in odd and even numbers" without us running out of time trying other methods? 
What is our plan A, B, and C as we are reading this problem? TY

---
Thus, let's play around!
99 is divisible by 11
100 - 1 is divisible by 11
10^2 - 1 is divisible by 11 : 99=100−1=102−199=100−1=102−1 : 10-raised to even power - 1

Similarly, we can see the next big number is of the format:
9999 is divisible by 11
10000 - 1 is divisible by 11
10^4 - 1 is divisible by 11 : 9999=10000−1=104−19999=10000−1=104−1 : 10-raised to even power - 1

MathRevolution · Answer

10^{98}  - 1

Start with lower power:  10^{2}  - 1 = 100 - 1 = 99

=>  \frac{99 }{ 11}  =   remainder is 0

=>  10^{3}  - 1 = 1000 - 1 = 999

=>  \frac{999 }{ 11}  =   remainder is 9

=>  10^{4}  - 1 = 10000 - 1 = 9999

=>  \frac{9999 }{ 11}  =   remainder is 0

Thus for the even power the remainder will be '0'.

Hence, =>  \frac{(10^{98} - 1) }{ 11}  =   remainder is 0

Answer A

bumpbot · Answer

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What is the remainder when 10^98-1 is divided by 11 ?

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What is the remainder when 10^98-1 is divided by 11 ?

Prep Toolkit

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