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# What is the remainder when 10^49 + 2 is divided by 11?

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Math Expert
Joined: 02 Sep 2009
Posts: 50624
What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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30 Jan 2015, 06:24
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24
00:00

Difficulty:

35% (medium)

Question Stats:

60% (00:52) correct 40% (00:44) wrong based on 518 sessions

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What is the remainder when $$10^{49} + 2$$ is divided by 11?

A. 1
B. 2
C. 3
D. 5
E. 7

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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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30 Jan 2015, 08:42
3
3
Hello Bunuel, I gave it a try:

I tried to find a pattern of remainder of (10 at any power) + 2 as following:

10^1 + 2 = 12/11 = 1 x 11 remainder 1
10^2 + 2 = 100 + 2 = 9 x 11 remainder 3
10^3 + 2 = 1000 + 2 = 91 x11 remainder 1
10^4 + 2 = 10000 + 2 = 909 x 11 remainder 3

It seems that any off power of 10, + 2 will results in a reminder = 1

Bunuel wrote:
What is the remainder when 10^49 + 2 is divided by 11?

A. 1
B. 2
C. 3
D. 5
E. 7

Kudos for a correct solution.

The OA will be revealed on Sunday

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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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30 Jan 2015, 08:10
Going by the logic
When 10^2/11 reminder is 1,
Similarly 10^3/11 reminder will be 1
and this would just continue regardless of 10 power anything (10^x)/11

just add 2 to this reminder, and hence the answer is 3.
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Joined: 02 Aug 2009
Posts: 7036
Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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30 Jan 2015, 08:54
1
2
Bunuel wrote:
What is the remainder when 10^49 + 2 is divided by 11?

A. 1
B. 2
C. 3
D. 5
E. 7

Kudos for a correct solution.

The OA will be revealed on Sunday

the property of 11 as divisor is (sum of odd digits -sum of even digits)=0 starting from rightmost digit..
this means for 10^even number, (sum of odd digits -sum of even digits) =1-0=1 so 1 is the remainder...
and for 10^odd number, (sum of odd digits -sum of even digits) =0-1=-1 so 10 is the remainder...
therefore 10^49 will have a remainder of 10.. so ans =12/11 ie rem=1 ans A
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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30 Jan 2015, 12:12
1
Hi quitdreaming,

You have to be very careful with your generalizations. If you're "off", even a little bit, then you'll likely get whatever question you're working on wrong.

With this calculation, you ARE correct:

100/11 = 9r1

But with this one, your generalization is incorrect:

1,000/11 = 90r10 (NOT remainder 1)

Knowing this, what would you do differently to solve this problem?

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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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Updated on: 31 Jan 2015, 06:38
3
10^49 + 2 / 11

--> +2 (11-1)^49 / 11

--> +2 -1 = 1

Originally posted by LaxAvenger on 30 Jan 2015, 14:38.
Last edited by LaxAvenger on 31 Jan 2015, 06:38, edited 1 time in total.
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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31 Jan 2015, 00:13
1
Bunuel wrote:
What is the remainder when 10^49 + 2 is divided by 11?

A. 1
B. 2
C. 3
D. 5
E. 7

Kudos for a correct solution.

The OA will be revealed on Sunday

Answer is A. $$10^odd$$+2 = Remainder 1

I tested the following cases$$10^2$$, $$10^3$$ and $$10^5$$ + 2
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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31 Jan 2015, 01:00
1
when 10^odd/11 reminder is 10; when 10^even/11 reminder is 1.

R10+R2 = R12. Since R 12 is not acceptable because it is greater than the divisor. R=1

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Math Expert
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Posts: 50624
Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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02 Feb 2015, 03:26
Bunuel wrote:
What is the remainder when 10^49 + 2 is divided by 11?

A. 1
B. 2
C. 3
D. 5
E. 7

Kudos for a correct solution.

The OA will be revealed on Sunday

VERITAS PREP OFFICIAL SOLUTION:

With exponent-based problems and huge numbers, it's often helpful to try to establish a pattern using small numbers. On this problem, while 1049+2 is a massive number that you'd never want to try to perform calculations with, you can start by using smaller numbers to get a feel for what it would look like:

10^2+2=102, which when divided by 11 produces a remainder of 3 (as 9 * 11 = 99, leaving 3 left over)

10^3+2=1002, which when divided by 11 produces a remainder of 1 (as 9 * 110 = 990, and when you add one more 11 to that you get to 1001, leaving one left)

10^4+2=10002, which when divided by 11 produces a remainder of 3 (as you can get to 9999 as a multiple of 11, which would leave 3 left over)

10^5+2=100002, which when divided by 11 produces a remainder of 1 (as you can get to 99990 and then add 11 more, bringing you to 100001 leaving one left over).

By this point, you should see that the pattern will repeat, meaning that when 10 has an even exponent the remainder is 3 and when it has an odd exponent the remainder is 1. Therefore, since 10 has an odd exponent in the problem, the remainder will be 1.
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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04 Feb 2015, 02:29
I solved it the following way:

1.10^49+2/11=10^51/11
2. Now i test cases 100/9 ==> Gives remainder of 1
3. If you know realize that the nominator has 3 letters you can deduct that 10^51/11 will also have a remainder of 1.
4. Counterproof 1000/11 ==> Remainder of 9

Bunuel wrote:
What is the remainder when 10^49 + 2 is divided by 11?

A. 1
B. 2
C. 3
D. 5
E. 7

Kudos for a correct solution.

The OA will be revealed on Sunday
Intern
Joined: 08 Dec 2013
Posts: 33
Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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15 Mar 2015, 18:39
what is the problem in solving indivitually like this..

10^49/11 + 2/11
from first part..we get 1 (odd places are 0 and even places are 9, as per cyclicity)
from second part, we get 2

1+2=3 =rem..
kindly correct me (we have used this line of method in example 1 also in math remainders theory..so wats the difference)

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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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15 Mar 2015, 18:54
1
shreygupta3192 wrote:
what is the problem in solving indivitually like this..

10^49/11 + 2/11
from first part..we get 1 (odd places are 0 and even places are 9, as per cyclicity)
from second part, we get 2

1+2=3 =rem..
kindly correct me (we have used this line of method in example 1 also in math remainders theory..so wats the difference)

hi shreygupta3192,
your approach is correct and can be done the way you have solved...
however you have to be careful while finding remainder..
when you divide 10^n by 11, the remainder will be 1 or -1, that is 10..... it will depend on n..
when you divide 10 by 11.. remainder is 10 and not 1... you add odd and even from right most digit and then subtract even from odd..
in case of 10... 0-1=-1or 10.... similarily 10^49 will give a remainder of 10..
overall remainder=10+2=12, which will give remainder as 1 when divided by 11..
hope it is clear..

chetan
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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20 Jan 2017, 08:57
Bunuel,
Can we solve this question using the below approach ?

10^49+2
=(11-1)^49+2
=11^49 - 1^49+2
=11^49 -1+2
=11^49 +1

Hence, when 10^49+2 is divided by 11 , implies when 11^49 +1 is divided 11, 11^49 is divisible by 11 leaving remainder 1 (due to +1 component).
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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21 Jan 2017, 03:20
grichagupta wrote:
Bunuel,
Can we solve this question using the below approach ?

10^49+2
=(11-1)^49+2
=11^49 - 1^49+2
=11^49 -1+2
=11^49 +1

Hence, when 10^49+2 is divided by 11 , implies when 11^49 +1 is divided 11, 11^49 is divisible by 11 leaving remainder 1 (due to +1 component).

No, that's wrong. (11-1)^49 does not equal to 11^49 -1^49. Does (a-b)^2 equals to a^2 - b^2?
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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21 Jan 2017, 03:38
Bunuel wrote:
grichagupta wrote:
Bunuel,
Can we solve this question using the below approach ?

10^49+2
=(11-1)^49+2
=11^49 - 1^49+2
=11^49 -1+2
=11^49 +1

Hence, when 10^49+2 is divided by 11 , implies when 11^49 +1 is divided 11, 11^49 is divisible by 11 leaving remainder 1 (due to +1 component).

No, that's wrong. (11-1)^49 does not equal to 11^49 -1^49. Does (a-b)^2 equals to a^2 - b^2?

Bunuel : Thank You. How silly of me ! I did some reading on this concept and now understand it. Thank you once again.
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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21 Jan 2017, 03:44
grichagupta wrote:
Bunuel wrote:
grichagupta wrote:
Bunuel,
Can we solve this question using the below approach ?

10^49+2
=(11-1)^49+2
=11^49 - 1^49+2
=11^49 -1+2
=11^49 +1

Hence, when 10^49+2 is divided by 11 , implies when 11^49 +1 is divided 11, 11^49 is divisible by 11 leaving remainder 1 (due to +1 component).

No, that's wrong. (11-1)^49 does not equal to 11^49 -1^49. Does (a-b)^2 equals to a^2 - b^2?

Bunuel : Thank You. How silly of me ! I did some reading on this concept and now understand it. Thank you once again.

Check the following post. It should help you in preparation: ALL YOU NEED FOR QUANT ! ! !.
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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21 Jan 2017, 11:49
10^2+2=102 and 102/11 Reminder is 3
10^3+2=1002 and now Reminder is 1
10^4+2=10002 Reminder is 3

So for 10^odd Reminder is 1 and for 10^even reminder is 3

As per the question we need reminder for 10^49 I.e. 10^odd

Hence and is A.(1)

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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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21 Jan 2017, 11:52
[quote="Vianand4"]10^2+2=102 and 102/11 Reminder is 3
10^3+2=1002 and now Reminder is 1
10^4+2=10002 Reminder is 3

So for 10^odd Reminder is 1 and for 10^even reminder is 3

As per the question we need reminder for 10^49 I.e. 10^odd

Hence and is A.(1)

Please correct me if I am wrong.

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What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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19 Sep 2017, 03:24
Rem|10/11| = -1

Rem|10^49/11| = (-1)^49 = -1

The FINAL remainder is :

Rem|(10^49 + 2)/11| = -1 + 2 = 1
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Re: What is the remainder when 10^49 + 2 is divided by 11?  [#permalink]

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19 Sep 2017, 04:05
Remainder of (10^49+2) when divided by 11 would be

=Remainder of 10^49 when divided by 11 + Remainder of 2 when divided by 11
=R (-1)^49/11 + 2/11
= -1+2
=1
Re: What is the remainder when 10^49 + 2 is divided by 11? &nbs [#permalink] 19 Sep 2017, 04:05

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