Question

When the positive integer $N$ is divided by $5$, the remainder is $3$. What is the remainder when $20N$ is divided by $25$?

Answer 1

Hint: When a natural number $a$ divided by a $q$,
$a = qx + r$ 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
$a$ 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 $5$ and
The value if $r$ is $3$
Therefore $N$ can be written as
$N = 5x + 3$
To find the remainder when $20N$ is divided by $25$. First, we are going to multiply $20$ in both sides of the above equation
Then we will get
$20N = 20(5x + 3)$
Further solving we get $20N = 100x + 60$
We have to take $25$ in common from the above equation. For that, I am writing it as
$20N = 100x + 50 + 10 \Rightarrow 20N = 25(4x + 2) + 10$ and clearly $25$ be divided and will be the quotient as $4x + 2$ and we will get the remainder as $10$ which is not divisible by the number $25$
Hence, we get the remainder when $20N$ is divided by $25$ is $10$
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 $y = 4x + 2$, because of $x$ is given that natural number, $4$and $2$ are natural numbers, $4x$ is a natural number (multiplication of two natural number is a natural number) then $4x + 2$ 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 integers values of the required question it is basically the highest common factor of the two numbers