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
