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When integer r is divided by 7, the remainder is 3. When

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aeros232
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When integer r is divided by 7, the remainder is 3. When [#permalink] New post   Updated on: 30 Sep 2012, 04:39
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When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 - s is divided by 7?

A. 2
B. 3
C. 4
D. 5
E. 6
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E

Originally posted by aeros232 on 29 Sep 2012, 13:14.
Last edited by Bunuel on 30 Sep 2012, 04:39, edited 2 times in total.
Renamed the topic and edited the question.
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EvaJager
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Re: What is the remainder when r^2 < s is divided by 7? [#permalink] New post   29 Sep 2012, 14:50
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aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 < s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


It should be \(r^2-s\) and not \(r^2<s\).

\(r=7a+3\) for some integer \(a\) and \(s=7b+3\) for some integer \(b.\)
Then \(r^2\) is a multiple M7 + 2 (multiple of 7 plus 2), so the difference \(r^2-s\) is M7 + 2 - 3 = M7 - 1 = M7 - 7 + 6 which is M7 + 6.

Answer E.
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Re: What is the remainder when r^2 < s is divided by 7? [#permalink] New post   29 Sep 2012, 15:01
EvaJager wrote:
aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 < s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


It should be \(r^2-s\) and not \(r^2<s\).

\(r=7a+3\) for some integer \(a\) and \(s=7b+3\) for some integer \(b.\)
Then \(r^2\) is a multiple M7 + 2 (multiple of 7 plus 2), so the difference \(r^2-s\) is M7 + 2 - 3 = M7 - 1 = M7 - 7 + 6 which is M7 + 6.

Answer E.


Yes, you are right!!! And by the way, excellent explanation, thanks!!
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Re: What is the remainder when r^2 - s is divided by 7? [#permalink] New post   29 Sep 2012, 23:52
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aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 - s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


i would suggest to take num rather than doing it algerabically.

take 10.. when it divided by 7 it will have remainder 3..
So R=10..

S will also B 10..

Square of 10..=100

r2-s..100-10=90..

90 /7 ..u l have remained 6 ..

Hope it helps !
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Re: What is the remainder when r^2 - s is divided by 7? [#permalink] New post   30 Sep 2012, 00:06
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aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 - s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


I was wondering if my approach is okay.. Here's my work.
r/7 = 3
s/7 = 3
r^2 - s / 7
Since everything is divided by 7, you can cancel all of those out.
R = S
R = 3
S = 3

so.. 3^2 - 3 = 6
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Re: What is the remainder when r^2 - s is divided by 7? [#permalink] New post   01 Oct 2012, 06:47
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sanjoo wrote:
aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 - s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


i would suggest to take num rather than doing it algerabically.

take 10.. when it divided by 7 it will have remainder 3..
So R=10..

S will also B 10..

Square of 10..=100

r2-s..100-10=90..

90 /7 ..u l have remained 6 ..

Hope it helps !


That's correct. +1.

r^2 - s must yield the same reminder when divided by 7 for every r and s which satisfy the given conditions. Thus picking numbers would be the easiest way to deal with this question.
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Re: When integer r is divided by 7, the remainder is 3. When [#permalink] New post   01 Oct 2012, 09:23
Bunuel wrote:
sanjoo wrote:
aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 - s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


i would suggest to take num rather than doing it algerabically.

take 10.. when it divided by 7 it will have remainder 3..
So R=10..

S will also B 10..

Square of 10..=100

r2-s..100-10=90..

90 /7 ..u l have remained 6 ..

Hope it helps !


That's correct. +1.

r^2 - s must yield the same reminder when divided by 7 for every r and s which satisfy the given conditions. Thus picking numbers would be the easiest way to deal with this question.



Thanks for kudos ..MATH GURU..:)..
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Re: When integer r is divided by 7, the remainder is 3. When [#permalink] New post   26 Oct 2012, 13:05
Or alternatively
r = 7x + 3
s = 7y + 3
r^2 = 49x^2+ 7*3*2 + 9
r^2 - s = 7*7x^2+ 7*3*2 + 9 - 7y + 3 = 7(7x^2+6) +6 = 7Q + 6
Remainder is 6
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When integer r is divided by 7, the remainder is 3. When [#permalink] New post   08 Dec 2017, 13:03
Bunuel wrote:
r^2 - s must yield the same reminder when divided by 7 for every r and s which satisfy the given conditions. Thus picking numbers would be the easiest way to deal with this question.


this doesn't seem to work when r=3 and s=10
not sure why
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Re: When integer r is divided by 7, the remainder is 3. When [#permalink] New post   08 Dec 2017, 22:44
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gracie wrote:
Bunuel wrote:
r^2 - s must yield the same reminder when divided by 7 for every r and s which satisfy the given conditions. Thus picking numbers would be the easiest way to deal with this question.


this doesn't seem to work when r=3 and s=10
not sure why


If r = 3 and s = 10, then r^2 - s = -1. The remainder when -1 id divided by 7 is also 6.
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Re: When integer r is divided by 7, the remainder is 3. When [#permalink] New post   21 Jun 2019, 20:36
Since we have absolute values in answer, plugged in 10,17 so 100 - 17 = 83, reminder 6
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When integer r is divided by 7, the remainder is 3. When [#permalink] New post   29 Aug 2023, 04:10
sanjoo wrote:
aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 - s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


i would suggest to take num rather than doing it algerabically.

take 10.. when it divided by 7 it will have remainder 3..
So R=10..

S will also B 10..

Square of 10..=100

r2-s..100-10=90..

90 /7 ..u l have remained 6 ..

Hope it helps !


Hi Sanjoo,

In this method, how do we know if R=S?
If R=10, S can be 3 or 10 or 17 etc, isn't it?
If S=3, then r2-s=100-9=91, 91/7 gives a remainder of 0.
If S=10, then as per the calculation, we get a remainder of 6.
Wondering if I missed something here?
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Re: When integer r is divided by 7, the remainder is 3. When [#permalink] New post   29 Aug 2023, 07:29
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pdavulur wrote:
sanjoo wrote:
aeros232 wrote:
52. When integer r is divided by 7, the remainder is 3. When integer s is divided by 7, the remainder is 3 also. What is the remainder when r^2 - s is divided by 7?

Show Spoiler
e


a. 2
b. 3
c. 4
d. 5
e. 6


i would suggest to take num rather than doing it algerabically.

take 10.. when it divided by 7 it will have remainder 3..
So R=10..

S will also B 10..

Square of 10..=100

r2-s..100-10=90..

90 /7 ..u l have remained 6 ..

Hope it helps !


Hi Sanjoo,

In this method, how do we know if R=S?
If R=10, S can be 3 or 10 or 17 etc, isn't it?
If S=3, then r2-s=100-9=91, 91/7 gives a remainder of 0.
If S=10, then as per the calculation, we get a remainder of 6.
Wondering if I missed something here?


Hi pdavulur,
Please check the highlighted part.
You have subtracted \(r^2-s^2 \)instead of \(r^2-s\)
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Re: When integer r is divided by 7, the remainder is 3. When [#permalink] New post   29 Aug 2023, 08:25
Thanks Nabneet for calling it out :)

I definitely missed that!
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Re: When integer r is divided by 7, the remainder is 3. When [#permalink]
29 Aug 2023, 08:25
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