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

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Woody19
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What is the remainder when 10^98-1 is divided by 11 ? [#permalink] New post   Updated on: 29 Jul 2019, 15:09
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Q. What is the remainder when (10^98 ) - 1 is divided by 11 ?

A. 0

B. 9

C. 11

D. 22

E. 33
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A

Originally posted by Woody19 on 29 Jul 2019, 11:41.
Last edited by Woody19 on 29 Jul 2019, 15:09, edited 1 time in total.
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Re: What is the remainder when 10^98-1 is divided by 11 ? [#permalink] New post   29 Jul 2019, 12:59
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Woody19 wrote:
Q. What is the remainder when (10^98 ) - 1 is divided by 11 ?

A. 0

B. 9

C. 11

D. 22

E. 33

I am sorry. I do not have answer for this question.

Need your help on this problem.


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! :grin:

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!
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Re: What is the remainder when 10^98-1 is divided by 11 ? [#permalink] New post   29 Jul 2019, 12:02
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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
Woody19 wrote:
Q. What is the remainder when (10^98 ) - 1 is divided by 11 ?

A. 0

B. 9

C. 11

D. 22

E. 33

I am sorry. I do not have answer for this question.

Need your help on this problem.


Posted from my mobile device
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Re: What is the remainder when 10^98-1 is divided by 11 ? [#permalink] New post   25 Sep 2020, 05:56
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\(10^{98}\) - 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
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Re: What is the remainder when 10^98-1 is divided by 11 ? [#permalink] New post   29 Jul 2019, 15:11
Staphyk

Xylan

Understood now :)

Thank you. +1
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Re: What is the remainder when 10^98-1 is divided by 11 ? [#permalink] New post   24 Sep 2020, 16:19
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
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Re: What is the remainder when 10^98-1 is divided by 11 ? [#permalink]
24 Sep 2020, 16:19
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