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If \(p\) is a positive integer, what is the remainder when \(p^2\) is divided by 12? (1) \(p\) is greater than 3. (2) \(p\) is a prime.
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27 Apr 2016, 00:54
Official Solution:
(1) \(p\) is greater than 3. Clearly insufficient: different values of \(p\) will give different values of the remainder. (2) \(p\) is a prime. Also insufficient: if \(p=2\) then the remainder is 4 but if \(p=3\) then the remainder is 9. (1)+(2) You can proceed with number plugging and try several prime numbers greater than 3 to see that the remainder will always be 1 (for example try \(p=5\), \(p=7\), \(p=11\)). If you want to doublecheck this with algebra, you should apply the following property of the prime number: any prime number greater than 3 can be expressed either as \(p=6n+1\) or \(p=6n1\). If \(p=6n+1\), then \(p^2=36n^2+12n+1\) which gives remainder 1 when divided by 12; If \(p=6n1\), then \(p^2=36n^212n+1\) which also gives remainder 1 when divided by 12.
Answer: C



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10 Jul 2018, 07:12
chetan2u niks18 pushpitkc Abhishek009I was stumped a bit on this question by trusting my gut that I have infinite prime numbers greater than 3 and did not bother to square them to check for remainder after dividing by 12. Could prime factorization of 12 would have been of any use? I did not understand algebraic approach used by Bunuel
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Re: M2707
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10 Jul 2018, 07:32
adkikani wrote: chetan2u niks18 pushpitkc Abhishek009I was stumped a bit on this question by trusting my gut that I have infinite prime numbers greater than 3 and did not bother to square them to check for remainder after dividing by 12. Could prime factorization of 12 would have been of any use? I did not understand algebraic approach used by Bunuel The prime numbers greater than 3 are of the form 6k+1... Why.. Because... 6k is div by 2 and 3... 6k+1 not by 2 and 3.. chances of it to be prime 6k+2 by 2 6k +3 by 3 6k+4 by 2 6k+5 not by 2 and 3, so may be prime again but 6k+5=6k+61=6(k+1)1 6k+6 is again of the form 6(k+1) So only possibility is 6k+1 or 6k1 When you square them (6k+1)=^2=36k^2+12k+1... All terms have 12 except 1... So remainder is 1 OR Just pick up some primes 5...5^3=25=12*2+1.. R is 1 7..7^2=49=12*4+1...R is 1
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Re: M2707
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17 Sep 2018, 08:40
If p is a positive integer statement 1: p>3, then P2 can be 16, 17, 18 etc and the remainder can be many numbers. hence insufficient. Statement 2:P is prime, P2 can be 9,25,49 etc and hence remainder can be any number. let us combine 1 and 2 statements, then P2 can be 25,49,121 etc and the remainder can be 1 only. hence answer is C
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17 Sep 2018, 09:43
1) if p is 4 then remainder of \(p^2\) is 4, but if p is 5 then remainder is 1. INSUFFICIENT 2) if p is 2, remainder is 4, but is p is 3 then remainder is 3. INSUFFICIENT Now by combining both statements, P should be greater than 3 and is a prime number then in all the possible cases, remainder of \(p^2\) is always 1. So Answer is C
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Re: M2707
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23 Sep 2018, 09:32
Bunuel wrote: If \(p\) is a positive integer, what is the remainder when \(p^2\) is divided by 12?
(1) \(p\) is greater than 3.
(2) \(p\) is a prime. Made a silly mistake by not testing 2 or 3 for Statement #2.



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Re: M2707
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30 Sep 2018, 05:54
I messed up here by not continuing with the calculation. Because there are many prime numbers that are greater than 3, I assumed the remainder for them all would be different when divided by 12 but it is not so. 5, 7, 11, 13 and 59 all yield a remainder of 1. Keeping that in mind, I assume, it will be the same for any other numbers too. And here I thought it was an easy question!
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Re: M2707
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07 Sep 2019, 07:43
y4nkee wrote: Official Solution:
(1) \(p\) is greater than 3. Clearly insufficient: different values of \(p\) will give different values of the remainder. (2) \(p\) is a prime. Also insufficient: if \(p=2\) then the remainder is 4 but if \(p=3\) then the remainder is 9. (1)+(2) You can proceed with number plugging and try several prime numbers greater than 3 to see that the remainder will always be 1 (for example try \(p=5\), \(p=7\), \(p=11\)). If you want to doublecheck this with algebra, you should apply the following property of the prime number: any prime number greater than 3 can be expressed either as \(p=6n+1\) or \(p=6n1\). If \(p=6n+1\), then \(p^2=36n^2+12n+1\) which gives remainder 1 when divided by 12; If \(p=6n1\), then \(p^2=36n^212n+1\) which also gives remainder 1 when divided by 12.
Answer: C And if n=14? p=6n+1 > p=85 not prime



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Re: M2707
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08 Sep 2019, 03:13
Fuf wrote: y4nkee wrote: Official Solution:
(1) \(p\) is greater than 3. Clearly insufficient: different values of \(p\) will give different values of the remainder. (2) \(p\) is a prime. Also insufficient: if \(p=2\) then the remainder is 4 but if \(p=3\) then the remainder is 9. (1)+(2) You can proceed with number plugging and try several prime numbers greater than 3 to see that the remainder will always be 1 (for example try \(p=5\), \(p=7\), \(p=11\)). If you want to doublecheck this with algebra, you should apply the following property of the prime number: any prime number greater than 3 can be expressed either as \(p=6n+1\) or \(p=6n1\). If \(p=6n+1\), then \(p^2=36n^2+12n+1\) which gives remainder 1 when divided by 12; If \(p=6n1\), then \(p^2=36n^212n+1\) which also gives remainder 1 when divided by 12.
Answer: C And if n=14? p=6n+1 > p=85 not prime Any prime number p, which is greater than 3, could be expressed as \(p=6n+1\) or \(p=6n+5\) or \(p=6n1\), where n is an integer greater than 1.Any prime number p, which is greater than 3, when divided by 6 can only give the remainder of 1 or 5 (remainder cannot be 2 or 4 as in this case p would be even and the remainder cannot be 3 as in this case p would be divisible by 3). So, any prime number p, which is greater than 3, could be expressed as \(p=6n+1\) or \(p=6n+5\) or \(p=6n1\), where n is an integer greater than 1. But:Not all number which yield a remainder of 1 or 5 upon division by 6 are primes, so viseversa of the above property is not true. For example 25 yields the remainder of 1 upon division be 6 and it's not a prime number. Hope it's clear.
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