Question

When a positive integer $n$ is divided by $5$, the remainder is $2$. What is the remainder when the number $3n$ is divided by $5$?

Answer 1

Hint: When a natural number $a$ divided by $q$ and get remainder r, then we can express this statement as:
$a = qx + r$
This is called the Euclid division algorithm.
where,
 $q$ is the quotient
 $r$ is the remainder
$x$ can be any natural number
$n$ be a given natural number.
This method was known as Euclid's algorithm.

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 $5$ and
The value if $r$ is $2$
Therefore $n$ can be written as
$n = 5x + 2$
Now we are going to find the value of $3n$.
To find $3n$, first, we are going to multiply $3$ in both sides of the above equation
Then we will get
$3n = 3(5x + 2)$
By doing calculations we will get
$3n = 15x + 6$
We have to take $5$in common from the above equation. For that, I am writing it as
$3n = 5(3x) + 5 + 1$
Now we can take $5$ in common in the equation. By taking $5$ we can get
$3n = 5(3x + 1) + 1$
By letting $y = 3x + 1$we will get $a = qx + r$$ap = qxp + rp$
$3n = 5y + 1$
The above equation is of the form of $a = qx + r$.
Here,
 $a$ is $3n$,
$q$ is 5
$x$ is $y$ ($y$ is also a natural number) and
$r$ is $1$.
Therefore, the remainder $r$ is $1$.
Additional Information:
The addition and multiplication of any two natural numbers result in a natural number.
To prove $y$ is also a natural number.
Because in the equation $y = 3x + 1$, because of $x$ is given that natural number, $3$ and $1$ are
natural numbers $3x$ is a natural number (multiplication of two natural numbers is a natural number) then $3x + 1$ 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.
For any positive integer $a$, we have seen that $a = qx + r$,
Whenever we are multiplying a natural number $p$ in the equation, then
$ap = qxp + rp$.
Hence the given question by putting the required values and simplifying according to the question.
Euclid’s algorithm is used for calculating positive integers values of the required question it is basically the highest common factor of the two numbers