Question

A number when divided by $342$ given a remainder $47$. When the same number is divide by $19$. what would be the remainder?

Answer 1

Let $n$ be a number divided by $342$ gives a remainder $47$.
From division algorithm, we have, Dividend $=($divisor$\times$ quotient)$+$ Remainder
Where, Dividend$=n$, divisor$=342$, quotient$=q$ is an integer and remainder $=47$
$\Rightarrow n=342q+47$.....(i)
Now to find remainder, when the same number $n$ divided by $19$,
We need to express RHS of equation(i) multiple of $19$.
$\Rightarrow n=(19\times 18q)+\left(\right[19\times 2]+9)$
$\Rightarrow n=19(18q+2)+9$....(ii)
Since, $q$ is an integer. So, $(18q+2)$ is also an integer.
Let $(18q+2)=k$ and substitute it in equation(ii), we get
$n=19k+9$
Applying the division algorithm, we get quotient $=k$ and remainder $=9$.
Therefore, the remainder when the number divided by $19$ is $9$.