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Math Expert V
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Question Stats: 71% (01:30) correct 29% (01:33) wrong based on 140 sessions

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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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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 double-check 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 $$p=6n-1$$.

If $$p=6n+1$$, then $$p^2=36n^2+12n+1$$ which gives remainder 1 when divided by 12;

If $$p=6n-1$$, then $$p^2=36n^2-12n+1$$ which also gives remainder 1 when divided by 12.

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chetan2u niks18 pushpitkc Abhishek009

I 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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chetan2u niks18 pushpitkc Abhishek009

I 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+6-1=6(k+1)-1
6k+6 is again of the form 6(k+1)

So only possibility is 6k+1 or 6k-1
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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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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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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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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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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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 double-check 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 $$p=6n-1$$.

If $$p=6n+1$$, then $$p^2=36n^2+12n+1$$ which gives remainder 1 when divided by 12;

If $$p=6n-1$$, then $$p^2=36n^2-12n+1$$ which also gives remainder 1 when divided by 12.

And if n=14?
p=6n+1 -> p=85 not prime
Math Expert V
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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 double-check 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 $$p=6n-1$$.

If $$p=6n+1$$, then $$p^2=36n^2+12n+1$$ which gives remainder 1 when divided by 12;

If $$p=6n-1$$, then $$p^2=36n^2-12n+1$$ which also gives remainder 1 when divided by 12.

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 $$p=6n-1$$, 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 $$p=6n-1$$, 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 vise-versa 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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