Question

When $12$ divided by a positive integer $n$, the remainder is $\left( n-3 \right)$, which of the following is the possible value of $n$?
A. $7$
B. $8$
C. $11$
D. $5$

Answer 1

Hint: In this problem we need to calculate the possible value for the given conditions. In the question we have that when $12$ is divided by a positive integer $n$, we will get $\left( n-3 \right)$ as a reminder. Here we will assume a constant $k$ as quotient. Now we will apply the division formula which is $\text{Dividend}=\text{Quotient}\times \text{Divisor}+\text{Reminder}$. After applying and simplifying the above rule we will get an equation or relation in terms of $n$, $k$. We will verify each pair of values which satisfies the obtained condition as well as the given condition. From those values we can choose the possible value of $n$.

Complete step by step solution:
Given that, when $12$ divided by a positive integer $n$, the remainder is $\left( n-3 \right)$.
Here Dividend is $12$, Divisor is $n$ and reminder is $\left( n-3 \right)$.
Let us assume the quotient as $k$.
We have the division rule as $\text{Dividend}=\text{Quotient}\times \text{Divisor}+\text{Reminder}$. Applying this the rule for the above data, then we will get
$12=k\times n+\left( n-3 \right)$
Simplifying the above equation by taking $n$ as common in left hand side, then we will get
$12=n\left( k+1 \right)-3$
Adding $3$ on both sides of the above equation, then we will get
$15=n\left( k+1 \right)$
We have the factors of the number $15$ as $1$, $3$, $5$, $15$.
All the above values we can write the values of $n$ as $1$, $3$, $5$, $15$.
From the given option we can choose the possible value of $n$ as $5$.

So, the correct answer is “Option D”.

Note: For this problem we can also check whether the given condition is satisfied by all the given options. First consider the value of $n$ as $7$ which is our first option. Now divide the number $12$ with $7$ and check whether the obtained remainder is equal to the value $\left( n-3 \right)$. If the remainder is equal to $\left( n-3 \right)$ then it’s our required option. We will check this for all options and find the correct option.