Question

Calculate the remainder when \[30\] is divided by \[7\]?

Answer 1

Hint: In order to find the remainder when \[30\] is divided by \[7\], we can either perform the process of division or apply the Euclid’s division algorithm and solve it by considering the remainder as a variable and the quotient as a nearer number that \[7\] divides. The value of the variable will be our required answer.

Complete step-by-step solution:
Now let us briefly talk about the Euclid division algorithm. It is also called as Euclid division Lemma which states that \[a,b\] are positive integers, then there exists unique integers satisfying \[q,r\] satisfying \[a=bq+r\] where \[0\le r< b\].
Now let us find the remainder when \[30\] is divided by \[7\].
We know that the Euclid division algorithm is \[a=bq+r\].
Here, we have \[a=30,b=7\].
In order to find \[q\], let us check such a number that divides \[30\] or nearly divides it.
So we can observe such number as \[28=7\times 4\]
We get the value of \[q\] as \[4\].
Upon substituting, we obtain
\[\begin{align}
  & \Rightarrow a=bq+r \\
 & \Rightarrow 30=7\left( 4 \right)+r \\
 & \Rightarrow 30=28+r \\
 & \Rightarrow 30-28=r \\
 & \Rightarrow r=2 \\
\end{align}\]
\[\therefore \] The remainder when \[30\] is divided by \[7\]is \[2\].

Note: We can also find the remainder by applying the division method as shown below.
\[\dfrac{30}{7}=4\]
Even in this case, we obtain the remainder as \[2\].
Using the Euclid division algorithm we can also find the HCF of the numbers. We must have a point to note that the numbers must be positive in order to apply the Euclid division algorithm in order to obtain a unique quotient and remainder.