Question

For any integers, a and 3, there exist unique integers q and r such that \[a=3q+r\] . Find the possible value of r.

Answer 1

Hint: We know the formula, \[\text{Dividend=Divisor }\!\!\times\!\!\text{ quotient+Remainder}\] . Now, compare the formula and the equation, \[a=3q+r\] . Then, get the values of the dividend, divisor, quotient, and the remainder. We know that when a number is divided by 3 then the possible values of r can be any of 0, 1, and 2. Now, get the possible values of r.

Complete step-by-step answer:
According to the question, it is given that for any integers, a and 3, there exist unique integers q and r such that \[a=3q+r\] . We have to find the possible values of r.
\[a=3q+r\] …………………………(1)
We know the formula, \[\text{Dividend=Divisor }\!\!\times\!\!\text{ quotient+Remainder}\] …………………………(2)
The dividend is the number that is getting divided. The divisor is the number which is dividing the dividend. The quotient is the result that is obtained when the dividend is divided by the divisor. The remainder is an integer that is left after dividing the dividend by the divisor to obtain an integer quotient.
Now, we can observe that (1) and (2) are identical.
On comparing, equation (1) and (2), we get
Dividend = a …………………….(3)
Divisor = 3 ………………………….(4)
Quotient = q …………………………(5)
Remainder = r …………………………..(6)
We can say that the integer ‘a’ is divided by 3 leaves ‘q’ as the quotient and ‘r’ as the remainder.
We know that when a number is divided by 3 then the possible values of the remainder can be any of 0, 1, and 2.
From equation (6), we have the ‘r’ as the remainder when the number ‘a’ is divided by 3.
Therefore, the possible values of r can be any of 0, 1, and 2.

Note: In this question, one might think to transform the given equation and then predict the values of r.
According, to the question, we have
\[a=3q+r\] ……………………(1)
Transforming equation (1), we get
\[\begin{align}
  & a=3q+r \\
 & \Rightarrow a-r=3q \\
\end{align}\]
\[\Rightarrow \dfrac{\left( a-r \right)}{3}=q\] ……………………….(2)
It is given that q is an integer, so \[\left( a-r \right)\] must be divisible by 3.
Now solving equation (2), we get
\[\dfrac{a}{3}-\dfrac{r}{3}=q\]
We know that the subtraction of two non-integral numbers can be an integer. So, it is not necessary that the number \[\dfrac{a}{3}\] and \[\dfrac{r}{3}\] must be an integer.
After this, we will not get any information about the values of ‘r’.
Hence, this is not the correct method to approach this question.