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If n and k are both positive integers, and n + 16 and n − 19 are both

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If n and k are both positive integers, and n + 16 and n − 19 are both [#permalink] New post   28 May 2019, 07:41
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If n and k are both positive integers, and n + 16 and n − 19 are both multiples of k, which of the following could be the value of k?

A. 3
B. 7
C. 16
D. 19
E. 70
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B
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If n and k are both positive integers, and n + 16 and n − 19 are both [#permalink] New post   28 May 2019, 07:50
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Starfruit wrote:
If n and k are both positive integers, and n + 16 and n − 19 are both multiples of k, which of the following could be the value of k?

A. 3
B. 7
C. 16
D. 19
E. 70


Since n + 16 and n − 19 are both multiples of k

(n+16)/k = x and (n-19)/k = y where x and y are integers

n = kx - 16 and n = ky + 19
—> kx - 16 = ky + 19
—> k(x - y) = 35
—> k = 35/(x - y)

Since, k is a positive integer. (x - y) should be a divisor of 35.

7 is the only option which is a divisor of 35.

IMO Option B.

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Re: If n and k are both positive integers, and n + 16 and n − 19 are both [#permalink] New post   28 May 2019, 12:22
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Starfruit wrote:
If n and k are both positive integers, and n + 16 and n − 19 are both multiples of k, which of the following could be the value of k?

A. 3
B. 7
C. 16
D. 19
E. 70


Number Property: If we add multiples of a number N, the result is a multiple of N.

Therefore, (n+16)+(n-19) is also a multiple of k
=> 2n-3 is a multiple of k

If we check the options, only option B is valid.


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Re: If n and k are both positive integers, and n + 16 and n − 19 are both [#permalink] New post   02 Jun 2019, 17:30
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Starfruit wrote:
If n and k are both positive integers, and n + 16 and n − 19 are both multiples of k, which of the following could be the value of k?

A. 3
B. 7
C. 16
D. 19
E. 70


Since the difference of n + 16 and n - 19 is (n + 16) - (n - 19) = 35, we see that k has to be a factor of 35. Of all the given answer choices, only 7 is a factor 35; thus it’s the correct answer:

Answer: B
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Re: If n and k are both positive integers, and n + 16 and n − 19 are both [#permalink] New post   29 Jan 2020, 16:21
Makeshift trick: Consider n = 40 so we get factors 21 and 56 .
Both of these are divisible by 7.

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Re: If n and k are both positive integers, and n + 16 and n 19 are both [#permalink] New post   06 Sep 2023, 23:42
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Starfruit wrote:
If n and k are both positive integers, and n + 16 and n − 19 are both multiples of k, which of the following could be the value of k?

A. 3
B. 7
C. 16
D. 19
E. 70


We can use this concept: If a, b are multiples of K, then the following will also be multiple of K;
1) a+b
2) a-b
3) ab
But same can't be said for when doing division.

Using this concept: n+16 and n-19 are multiple of K, then using (2) from above concept , \(n+16 - (n-19) = 35\). So we know 35 is multiple of k. Then all the factors of 35 will be factors of K too i.e 5 and 7.
option B is correct.
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Re: If n and k are both positive integers, and n + 16 and n 19 are both [#permalink]
06 Sep 2023, 23:42
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