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If p is a positive integer, is p even?

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If p is a positive integer, is p even?

by VJesus12 » Mon May 21, 2018 7:33 am

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If p is a positive integer, is p even?

(1) p divided by 3 leaves a remainder of 1
(2) p divided by 4 leaves a remainder of 1

The OA is the option B .

Why is sufficient the second statement? How can I get an answer? Help!!!
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by Brent@GMATPrepNow » Mon May 21, 2018 7:42 am

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VJesus12 wrote:If p is a positive integer, is p even?

(1) p divided by 3 leaves a remainder of 1
(2) p divided by 4 leaves a remainder of 1
Target question: Is p even?

Statement 1: p divided by 3 leaves a remainder of 1
In other words, p is 1 greater than some multiple of 3
There are several values of p that satisfy this condition. Here are two:
Case a: p = 4. In this case, the answer to the target question is YES, p is even
Case b: p = 7. In this case, the answer to the target question is NO, p is not even
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: p divided by 4 leaves a remainder of 1
In other words, p is 1 greater than some multiple of 4
So, we can say that: p = 4k + 1 for some integer k
Since 4k must be EVEN, we can conclude that 4k+1 must be ODD
In other words, p MUST be ODD
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
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by Vincen » Mon May 21, 2018 11:43 pm

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VJesus12 wrote:If p is a positive integer, is p even?

(1) p divided by 3 leaves a remainder of 1
(2) p divided by 4 leaves a remainder of 1

The OA is the option B .

Why is sufficient the second statement? How can I get an answer? Help!!!
Hello Vjesus12.

Let's take a look.

We have to say if p is even or not.
(1) p divided by 3 leaves a remainder of 1
This implies that $$p=3\cdot k+1,\ \ \ \ k\in\mathbb{Z}.$$ Now,

1- If k=1 then p=4, that is to say, p is EVEN.
2- If k=2 then p=7, that is to say, p is ODD.

NOT SUFFICIENT.
(2) p divided by 4 leaves a remainder of 1
This implies that $$p=4\cdot k+1,\ \ \ \ k\in\mathbb{Z}.$$ Now, since 4k is always an even number, then 4k+1 will be always an ODD number. Hence, p is always ODD.

Therefore, the answer to the question "is p even?" is NO, it isn't.

SUFFICIENT.

So, the correct answer is the option B.

I hope it helps.
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by Brent@GMATPrepNow » Tue May 22, 2018 6:16 am

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VJesus12 wrote:If p is a positive integer, is p even?

(1) p divided by 3 leaves a remainder of 1
(2) p divided by 4 leaves a remainder of 1
Target question: Is p EVEN?

ASIDE: Another way to handle statement 2 is to apply a useful rule:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Statement 2: p divided by 4 leaves a remainder of 1
So, some possible values of p are: 1, 5, 9, 13, 17, 21, 25, 29, ...etc
We can see that, if we continue listing possible values of p, all of those values will be ODD
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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by Scott@TargetTestPrep » Thu May 24, 2018 12:35 pm

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VJesus12 wrote:If p is a positive integer, is p even?

(1) p divided by 3 leaves a remainder of 1
(2) p divided by 4 leaves a remainder of 1
We need to determine whether positive integer p is even.

Statement One Alone:

p divided by 3 leaves a remainder of 1

We see that p could be 4 or p could be 7. Statement one is not sufficient to answer the question.

Statement Two Alone:

p divided by 4 leaves a remainder of 1

See see that p is 1 more than any multiple of 4. Since all multiples of 4 are even, p must be odd. Statement two is sufficient to answer the question.

Answer: B

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