Question

A positive integer $n$ when divided by $8$ leaves a remainder $5$. What is the remainder when $2n + 4$ is divided by $8$?
$A)8$
$B)1$
$C)6$
$D)0$

Answer 1

When a natural number $a$ divided by a $q$ and leaves remainder r, as
$a = qx + r$ and this process will be called Euclid’s lemma of the division
where,
$q$ is the quotient
$r$ is the remainder
$x$ can be any natural number
$n$ be a given natural number.

Complete step-by-step solution:
Now we are going to find $n$ using the hint
From the problem we are given that
The value of $q$ is $8$ and
The value if $r$ is $5$
Therefore $n$ can be written as
$n = 8x + 5$
Now we are going to find the value of $2n + 4$.
To find $2n + 4$, first, we are going to multiply $2$ and add $4$ in both sides of the above equation
Then we will get
$2n + 4 = 2(8x + 5) + 4 \Rightarrow 16x + 14$
We have to take $8$ in common from the above equation. For that, I am writing it as
$2n + 4 = 16x + 14 \Rightarrow (2x + 1)8 + 6$ where $8$ is the quotient here.
Hence, we get the remainder when $2n + 4$ is divided by $8$ is $6$
Therefore, the option $C)6$ is correct.
Additional Information:
The addition and multiplication of any two natural numbers result in a natural number. To prove $x$is also a natural number. Because in the equation $y = 2x + 1$, because of $x$ is given that natural number, $3$and $1$ are natural numbers,$2x$ is a natural number (multiplication of two natural number is a natural number) then $2x + 1$ is also a natural number (Addition of two natural numbers is a natural number).

Note: To solve these types of the question, we should apply Euclid’s algorithm in the given statement. Then calculate the values of the dividend, divisor and quotient, and remainder. Hence the given question by putting the required values and simplifying according to the question. Euclid’s algorithm is used for calculating positive integer values of the required question; it is basically the highest common factor of the two numbers.