Properties Of Integers - OG

[shanice]

When 10 is divided by the positive integer n, the remainder is n-4. Which of the following could be the value of n?

(A) 3 (B)4 (c)7 (D)8 (E)12

Answer is C.

I arrived at this final equation 14=qn+n=n(q+1). I don't understand why the OG says that n must be a factor of 14 and so n=1,2,7 or 14 since n is a positive integer. The only positive integer factor of 14 given in the answer choices is 7.

Can someone please explain to me in detail , why n must be a factor of 14?

Please help. Thank you.

[neelgandham]

I haven't checked the OG solution but here is what I think.

When 10 is divided by the positive integer n, the remainder is n-4. Which of the following could be the value of n?

We know that remainder is always greater than 0. i.e., n-4 > 0, implies n>4. Now going by your explanation..

I arrived at this final equation 14=qn+n=n(q+1). I don't understand why the OG says that n must be a factor of 14 and so n=1,2,7 or 14 since n is a positive integer. The only positive integer factor of 14 given in the answer choices is 7.

We can, now, eliminate the values n=1,2 as n(1)-1,n(2)-1 <4. so n = 7 or 14 and is always a factor of 14.

p.s: I will check the OG solution and will get back to you with a better explanation, if any.

[shanice]

Thanks for your fast response. But I still can't understand. Can you please explain further?

Thank you.

[neelgandham]

Hey Shanice,

No problem! I will try to explain it further to the best of my knowledge.

If a = b*c, where a,b,c are positive integers, then b,c are factors of a. Similarly, if 14 = n*(q+1), then n and (q+1) are factors of 14

Let me solve this for you!
The School way
Firstly, do you agree that remainder is a non-negative value? If the answer is NO please read the quoted

If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively, such that y = xq + r and 0 ≤ r < x. - Source: Official Guide-12 Page 108

Now that we agree that remainder is a non-negative value, we know n-4>0, which implies n>4. We also know that 14=qn+n=n(q+1), where q is the quotient and q>0.

14 = n(q+1)
14 = 1*14, 2*7, 7*2, 14*1 are the only possible combinations as n and q+1 are positive integers.

Case 1: 14 = 1*14 = n(q+1), n = 1 and q =13, but n-4= -3<0, which should not be the case, as n>4
Case 2: 14 = 2*7 = n(q+1), n = 2 and q =6, but n-4= -3<0, which should not be the case, as n>4
Case 3: 14 = 14*1 = n(q+1), n = 14 and q =0, but n-4= 10>0, so n = 14 can be the answer
Case 4: 14 = 7*2 = n(q+1), n = 7 and q = 1, but n-4= 3>0, so n = 7 can be the answer

So, since n should be one among 1,2,7,14 Official Guide stated that n must be a factor of 14. I hope that answers your question.

Here is an easier and a quicker solution to this question, the GMAT way

When 10 is divided by the positive integer n, the remainder is n-4. Which of the following could be the value of n?

(A) 3, if n=3,the remainder = n-4 = -1, but when 10 is divided by 3 the remainder is 1. Incorrect!
(B) 4, if n=4, the remainder = n-4 = 0, but when 10 is divided by 4 the remainder is 2. Incorrect!
(c) 7, if n=7, the remainder = n-4 = 3, and when 10 is divided by 7 the remainder is 3. Correct!
(D) 8 - Don't bother
(E) 12 - Don't bother

[Bill@VeritasPrep]

Once you're at the equation 14=n(q+1), you know that n * (q + 1) gives you a product of 14. Since we know n must be an integer, it (and by extension, q + 1) must be a factor of 14.

[shanice]

A big thank you to neelgandham. I'm so happy as I could understand it now. Thank you for putting the effort to explain it to me in detail. I really appreciate it.

And also not forgetting, Mr.Smith. Thank you.

[f430spy]

10/n=q+(n-4)/n where q is an integer.

14=n(q+1)->(q+1)=14/n

Given that q+1 should be an integer, then n have to be a factor of 14.