Divide and check whether your answer by the corresponding multiplication of the following: \[612846\div 582\]?

Hint: From the question we have been asked to find the solution to \[612846\div 582\] and check that the answer we found is correct or not using the multiplication operation. So, for this question we will use the division operation in mathematics and simplify expression and find the solution to the question.

Complete step-by-step solution:
Firstly, from given,
We can rewrite the given question \[612846\div 582\] using the basic mathematical operation which is division as follows.
\[\Rightarrow \dfrac{612846}{582}\]
Now, we will simplify the above fraction so that it makes our solution much easier and reduced form by dividing both the numerator and denominator with the same number so that it does not change the fraction value.
So, as described in the above sentence now we will divide the numerator and denominator with the same number.
\[\Rightarrow \dfrac{\dfrac{612846}{2}}{\dfrac{582}{2}}\]
\[\Rightarrow \dfrac{306423}{291}\]
Now, we write prime factorisation of the numerator and denominator and cancel common terms.
\[\Rightarrow \dfrac{{{3}^{5}}\times 13\times 97}{3\times 97}\]
\[\Rightarrow {{3}^{4}}\times 13\]
\[\Rightarrow 1053\]
Here, we got the answer as \[\Rightarrow 1053\] when we divide the \[612846\] with the number \[582\].
Now, we will check the answer by using multiplication operations.
We know that \[\text{dividend}=\text{divisor}\times \text{quotient+remainder}\]. Here we did not get any remainder. So, the remainder is zero. From the question we are given that the dividend is \[612846\] and also the divisor as \[582\].
We have found out that the quotient is as follows.
\[\Rightarrow 1053\]
So, now we will check the solution by substituting the values in the \[\text{dividend}=\text{divisor}\times \text{quotient+remainder}\] rule. So, the equation will be reduced as follows.
\[\Rightarrow 612846=582\times 1053+0\]
\[\Rightarrow 612846=582\times 1053\]
\[\Rightarrow 612846=612846\]
Here, LHS is equal to RHS. So, we can say our solution is correct.

Note: Students must be very careful in doing the calculations. Students must not be confused in using the formula \[\text{dividend}=\text{divisor}\times \text{quotient+remainder}\]. We should not do mistake in substituting the values wrong in this formula like if we write it as \[\Rightarrow 612846=582\times 0+1053\] instead of \[\Rightarrow 612846=582\times 1053+0\] then our solution will be wrong.