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QOTD#15 When m is divided by 11, the remainder is r.

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QOTD#15 When m is divided by 11, the remainder is r. [#permalink] New post 19 Aug 2016, 03:50
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56% (01:12) correct 44% (00:52) wrong based on 25 sessions
\(m = 10^{32} + 2\)

When m is divided by 11, the remainder is r.

Quantity A
Quantity B
r
3


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

Practice Questions
Question: 8
Page: 150
[Reveal] Spoiler: OA

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Re: QOTD#15 When m is divided by 11, the remainder is r. [#permalink] New post 19 Aug 2016, 03:55
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Explanation

Actually dividing \(10^3^2 + 2\) by 11 would be very time consuming, so it is worth trying to compare the quantities without actually doing the division.

A good approach would be to compute the remainders when \(10^1 + 2\), \(10^2 + 2\), \(10^3 + 2\), \(10^4 + 2\), etc., are divided by 11 to see if there is a pattern that can help you determine the remainder when \(10^3^2 + 2\) is divided by 11. The following table shows the first few cases.

Image

Note that the remainder is 1 when 10 is raised to an odd power, and the remainder is 3 when 10 is raised to an even power. This pattern suggests that since 32 is even, the remainder when \(10^3^2 + 2\) is divided by 11 is 3.

To see that this is true, note that the integers 99 and 9,999 in the rows for n = 2 and n = 4, respectively, are multiples of 11. That is because they each consist of an even number of consecutive digits of 9. Also, these multiples of 11 are each 3 less than \(10^2 + 2\) and \(10^4 + 2\), respectively, so that is why the remainders are 3 when \(10^2 + 2\) and \(10^4 + 2\) are divided by 11.

Similarly, for n = 32, the integer with 32 consecutive digits of 9 is a multiple of 11 because 32 is even. Also, that multiple of 11 is 3 less than \(10^3^2 + 2\), so the remainder is 3 when \(10^3^2 + 2\) is divided by 11. Thus the correct answer is Choice C.

[Reveal] Spoiler: Img
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Re: QOTD#15 When m is divided by 11, the remainder is r. [#permalink] New post 27 Aug 2016, 11:58
Hi Sandy,

I do get that 10^32 will have a remainder of 3 but what about (+2) which is in the question. Will it not effect the remainder that we get from 10^32? How does that work?

Thanks for your help!

HK
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Re: QOTD#15 When m is divided by 11, the remainder is r. [#permalink] New post 27 Aug 2016, 14:15
Expert's post
HarveyKlaus wrote:
Hi Sandy,

I do get that 10^32 will have a remainder of 3 but what about (+2) which is in the question. Will it not effect the remainder that we get from 10^32? How does that work?

Thanks for your help!

HK


Hey,

\(\frac{(10^3^2 +2 )}{11}\) has a remainder of 3 not \(\frac{10^3^2}{11}\).

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Re: QOTD#15 When m is divided by 11, the remainder is r. [#permalink] New post 23 Mar 2018, 12:47
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sandy wrote:
\(m = 10^{32} + 2\)

When m is divided by 11, the remainder is r.

Quantity A
Quantity B
r
3


A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

Practice Questions
Question: 8
Page: 150


m = 10^(32) + 2 = 100,000,000,000,000,000,000,000,000,000,002 so we certainly don't want to CALCULATE the remainder when m is divided by 11.
Instead, let's examine different powers of 10 and look for a pattern

1) 10^1 + 2 = 12. When we divide 12 by 11, we get remainder 1
2) 10^2 + 2 = 102. When we divide 102 by 11, we get remainder 3
3) 10^3 + 2 = 1002. When we divide 1002 by 11, we get remainder 1
4) 10^4 + 2 = 10,002. When we divide 10,002 by 11, we get remainder 3

So, when the exponent is ODD, the remainder is 1
And, when the exponent is EVEN, the remainder is 3

So, when we divide 10^32 +2 by 11, we get remainder 3

So, we have:
Quantity A: 3
Quantity B: 3

Answer: C

Cheers,
Brent
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Re: QOTD#15 When m is divided by 11, the remainder is r. [#permalink] New post 27 Nov 2018, 02:16
thank you very much
Re: QOTD#15 When m is divided by 11, the remainder is r.   [#permalink] 27 Nov 2018, 02:16
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QOTD#15 When m is divided by 11, the remainder is r.

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