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What is the largest prime number P so that (p^2)/12 leaves a

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Senior Manager
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What is the largest prime number P so that (p^2)/12 leaves a [#permalink] New post 17 Jul 2004, 06:00
What is the largest prime number P so that
(p^2)/12 leaves a remainder that is different from 1 ?

A) 2
B) 3
C) 7
D) 21
E) 97
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 [#permalink] New post 17 Jul 2004, 06:24
I guess from trial and error the ans is b 3

we can eliminate d 21 which isnt a prime . 97 can be expreesed as 96 + 1
so 97^2 will leave a remainder of 1 so alll that remain are a,b, c . 7 leaves a remainder of 1 again so between 2 and 3 its 3
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 [#permalink] New post 17 Jul 2004, 06:31
Agreed ans should be B :yes
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 [#permalink] New post 17 Jul 2004, 07:35
Why not 2 ?

2 divided by 12 leaves a remainder of 2 which is different from 1.

Why do I feel that the question has some discrepency?

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 [#permalink] New post 17 Jul 2004, 11:40
smandalika wrote:
I guess from trial and error the ans is b 3

we can eliminate d 21 which isnt a prime . 97 can be expreesed as 96 + 1
so 97^2 will leave a remainder of 1 so alll that remain are a,b, c . 7 leaves a remainder of 1 again so between 2 and 3 its 3


can you eloborate on your answer pls
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 [#permalink] New post 17 Jul 2004, 15:31
abisurd wrote:
Why not 2 ?

2 divided by 12 leaves a remainder of 2 which is different from 1.

Why do I feel that the question has some discrepency?


3 is larger than 2, and remainder in a case of 3 is not equal to one, hence you pick 3
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 [#permalink] New post 17 Jul 2004, 22:15
K_s_r wrote:
smandalika wrote:
I guess from trial and error the ans is b 3

we can eliminate d 21 which isnt a prime . 97 can be expreesed as 96 + 1
so 97^2 will leave a remainder of 1 so alll that remain are a,b, c . 7 leaves a remainder of 1 again so between 2 and 3 its 3


can you eloborate on your answer pls


Please explain your answer...:thanks
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 [#permalink] New post 18 Jul 2004, 07:12
We have to find LARGEST Prime number that leaves remainder ~= 1

Start from
(E) : 97 is of the form (12n + 1 ) will always give remainder 1
(D) : 21 forget it Not Prime.
(C) : 7 -> 49 Remainder 1
(B) : 3 -> 9 Remainder 9
Thats it : (B) is the answer.

No need to look for (A) because even if that would be answer, 3 will always be greater than 2.
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 [#permalink] New post 18 Jul 2004, 07:57
Assuming B is the final ans I will try to give a better explanation.


First off the question is about a prime number so 21 not being a prime is eliminated.

next 97 can be expressed as (96 + 1)^ 2 = 96^2 + 1 + 2*96*1
since (a + b)^2 = a^2 + b^2 + 2*a*b

so since 96 is divisible by 12 97^2/12 leaves a remainder of 1

7^2 / 12 = 49/12 leaves a remainder of 1 so the only 2 options left out are 2 and 3
2^2 / 12 leaves a remainder of 4 and
3^2/ 12 leaves a remainder of 9

but we are looking for the biggest prime number so its 3 between the 2 alternatives
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 [#permalink] New post 18 Jul 2004, 08:33
great explanation!
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 [#permalink] New post 18 Jul 2004, 13:03
The OA is B.

As a matter of fact, every prime number greater than 3, when squared and then divided by 12,
leaves a remainder of 1.

Try it!
  [#permalink] 18 Jul 2004, 13:03
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