Question

When $12$ divided by a positive integer $n,$ the remainder is $(n-3),$ which of the following is the possible value of ${}^{\prime }{n}^{\prime }?$

• A. $7$
• B. $8$
• C. $11$
• D. $5$

Answer 1

The correct option is D $5$

When $12$ divided by a positive integer $n$, the remainder is $(n-3)$.
Thus, $12=nk+(n-3)$, where is a $k$ positive integer
$\Rightarrow 12=nk+n-3$
$\Rightarrow 15=n(k+1)$
The divisors of $15$ are $1,3,5,15.$
Thus, the possible values of $(n,k+1)$ are $(1,15),(15,1),(3,5),(5,3).$
Now, $(1,15),(15,1),(3,5)$ does not satisfy our initial equation $12=nk+(n-3).$
Therefore $n=5$ is the possible value.