Question

When a positive integer $m$ is divided by 7, the remainder is 4. What is the remainder when $2m$ is divided by 7?
A) 9
B) 1
C) 4
D) 6
E) 8

Answer 1

Hint: If dividend is double and divisor is same so remainder is also double but it should be less than divisor if it is greater than divisor then, that remainder should be again divided by divisor to get the original reminder.

Complete step by step solution:
Given $m$ is integer that is dividend,
Suppose $m=7x+r$
Where $r$ is remainder i.e. 7 and $x$is any whole number.
So,$m=7x+4$
Now multiply 2 both sides to get the value of $2m$
$\Rightarrow 2m=2\left( 7x+4 \right)$
$\Rightarrow 2m=14x+8$
So, now we divide $2m$by 7
$\Rightarrow \dfrac{2m}{7}=\dfrac{7y+{{r}_{2}}}{7}$
Where $y$is any whole number and ${{r}_{2}}$ be the remainder
$\Rightarrow \dfrac{14x+8}{7}=\dfrac{7y+{{r}_{2}}}{7}$
$\Rightarrow 7\left( \dfrac{2x}{7} \right)+\dfrac{8}{7}=\dfrac{7y+{{r}_{2}}}{7}$
$\Rightarrow 7\left( \dfrac{2x}{7} \right)+\dfrac{7+1}{7}=\dfrac{7y+{{r}_{2}}}{7}$
$\Rightarrow 7\left( \dfrac{2x+1}{7} \right)+\dfrac{1}{7}=\dfrac{7y+{{r}_{2}}}{7}$
$\Rightarrow 7\left( \dfrac{2x+1}{7} \right)+\dfrac{1}{7}=\dfrac{7y}{7}+\dfrac{{{r}_{2}}}{7}$
Now compare the equation both side and we get ${{r}_{2}}=1$
Then the remainder will be 1

Hence option (B) is correct.

Additional information:
In division we will see the relationship between the dividend, divisor, quotient and remainder. The number which we divide is called the dividend. The number by which we divide is called the divisor. The result obtained is called the quotient. The number left over is called the remainder.
Dividend = divisor × quotient + remainder

Note: Don’t confuse over dividend and divisor, doubled remainder of doubled dividend is not always a correct, it should be less than divisor.