Question

What is the remainder when $9875347\times 7435789\times 5789743$ is divided by 4?

Answer 1

Hint: We first try to find the modulo form of 4 for the given numbers $9875347,7435789,5789743$. The multiplication of the numbers gives the modulo form the final answer which in turn gives the remainder of the solution.

Complete step-by-step answer:
We form the given numbers with respect to the modulo form of 4.
We get
$\begin{align}
  & 9875347=4\times 2468836+3=4a+3 \\
 & 7435789=4\times 1858947+1=4b+1 \\
 & 5789743=4\times 1447435+3=4c+3 \\
\end{align}$
where $a,b,c\in \mathbb{N}$.
Therefore, the multiplication form becomes $\left( 4a+3 \right)\left( 4b+1 \right)\left( 4c+3 \right)$.
We try to find the remainder form of the multiplication.
$\begin{align}
  & \left( 4a+3 \right)\left( 4b+1 \right)\left( 4c+3 \right) \\
 & =\left( 16ab+4a+12b+3 \right)\left( 4c+3 \right) \\
 & =64abc+16ac+48bc+12c+48ab+12a+36b+9 \\
 & =4\left( 16abc+4ac+12bc+3c+12ab+3a+9b+2 \right)+1 \\
\end{align}$
Therefore, the multiplication of $9875347\times 7435789\times 5789743$ will be of the form $4n+1;n\in \mathbb{N}$.
The remainder will be 1.
So, the correct answer is “1”.

Note: We need to be careful about the quotient being integer. as we have taken $a,b,c\in \mathbb{N}$, the final quotient $16abc+4ac+12bc+3c+12ab+3a+9b+2\in \mathbb{N}$, as they are the multiplication form of the integers.