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A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a

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Bunuel
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A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   23 Apr 2018, 00:41
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A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a positive integer. What is the value of n?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 13
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DavidTutorexamPAL
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A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   Updated on: 23 Apr 2018, 04:39
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Bunuel wrote:
A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a positive integer. What is the value of n?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 13


Questions dealing with number properties, factorizations and remainders can often be solved with logic and very few calculations.
We'll look for such a solution, a Logical approach.

Since n is a factor of 7k+1 it must also be a factor of 2*(7k+1) = 14k+2.
Since n is a factor of both 14k+13 and 14k+2 it is also a factor of the difference between them, that is of (14k+13)-(14k+2) = 11.
Since 11 has only two factors, 1 and 11 and n is prime, n=11.

(D) is our answer.

Originally posted by DavidTutorexamPAL on 23 Apr 2018, 00:59.
Last edited by DavidTutorexamPAL on 23 Apr 2018, 04:39, edited 2 times in total.
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A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   23 Apr 2018, 04:00
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DavidTutorexamPAL wrote:
Bunuel wrote:
A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a positive integer. What is the value of n?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 13


Questions dealing with number properties, factorizations and remainders can often be solved with logic and very few calculations.
We'll look for such a solution, a Logical approach.

Since n is a factor of 7k+1 it must also be a factor of 2*(7k+1) = 14k+7.
Since n is a factor of both 14k+13 and 14k+7 it is also a factor of the difference between them, that is of (14k+13)-(14k+7) = 6.
Then n can be 1,2,3,6 but since it is prime it can only be 2 or 3.
To decide between 2 and 3 we'll need to see if have any even/odd properties.
14k is even and 13 is odd so 14k+13 must be odd. That is, n is a factor of an odd number and cannot be 2.

(B) is our answer.


Hi,

Isn't 2(7k+1) = 14k + 2? And therefore difference will be 11. I got 11 by substituting possible values for k.

Please correct me if I am wrong.


Thanks!
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Re: A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   23 Apr 2018, 04:37
urvashis09 wrote:
DavidTutorexamPAL wrote:
Bunuel wrote:
A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a positive integer. What is the value of n?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 13


Questions dealing with number properties, factorizations and remainders can often be solved with logic and very few calculations.
We'll look for such a solution, a Logical approach.

Since n is a factor of 7k+1 it must also be a factor of 2*(7k+1) = 14k+7.
Since n is a factor of both 14k+13 and 14k+7 it is also a factor of the difference between them, that is of (14k+13)-(14k+7) = 6.
Then n can be 1,2,3,6 but since it is prime it can only be 2 or 3.
To decide between 2 and 3 we'll need to see if have any even/odd properties.
14k is even and 13 is odd so 14k+13 must be odd. That is, n is a factor of an odd number and cannot be 2.

(B) is our answer.


Hi,

Isn't 2(7k+1) = 14k + 2? And therefore difference will be 11. I got 11 by substituting possible values for k.

Please correct me if I am wrong.


Thanks!


You're absolutely right!
Not sure how that 7 slipped in there....
Thanks for the correction, I'll edit the original post.
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Re: A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   01 May 2018, 05:27
Bunuel wrote:
A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a positive integer. What is the value of n?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 13


I'm sure this method is probably not correct, but I got the correct answer...

Add 14k+13 and 7k+1. =21k+14.

21k+14 is divisible by 7.

3k+2. Now we know k is a positive integer, so lets plug in positive integers and see if we can get a prime number for n.

k=1 -->n=5, which is prime, but is not an answer choice.
k=2 -->n=8, which is not prime.
k=3 -->n=11, which is prime and an answer choice.

Answer D, n=11
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A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   01 May 2018, 13:34
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Bunuel wrote:
A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a positive integer. What is the value of n?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 13


if n is a factor of 14k+13 and 7k+1,
then it must be a factor of (14k+13)-(7k+1), or 7k+12,
and thus a factor of (7k+12)-(7k+1), or 11
11
D
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Re: A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   03 May 2018, 16:08
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Bunuel wrote:
A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a positive integer. What is the value of n?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 13


If k = 1, then 14k + 13 = 27 and 7k + 1 = 8. The only factor that is common to 27 and 8 is 1, however, 1 is not in the choices.

If k = 2, then 14k + 13 = 41 and 7k + 1 = 15. The only factor that is common to 41 and 15 is 1, however, 1 is not in the choices.

If k = 3, then 14k + 13 = 55 and 7k + 1 = 22. The factors that are common to 55 and 22 are 1 and 11, and 11 is in the choices. So n = 11.

Answer: D
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Re: A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   09 May 2018, 07:28
Best approach is to multiply 2*(7k+1)
=14k +2

So factor would be difference of 14k+13 and 14k+2
I.e. 11

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Re: A prime number n is a factor of both 14k + 13 and 7k + 1, where k is a [#permalink] New post   11 May 2021, 01:32
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