Question

Let a be an integer and d be a positive integer. Show that the integers qand r with$\mathbf{a}\mathbf{=}\mathbf{dq}\mathbf{+}\mathbf{r}$ and$\mathbf{0}\mathbf{\le }\mathbf{r}\mathbf{<}\mathbf{d}$ which were shown to exist in Example 5, are unique.

Short answer

The integersr andQ are unique.

Step-by-step solution

Identification of the given data

The given data can be listed below as:

• The value of the first integer is a.
• The value of the positive integer is d.
• The value of the second integer is q.
• The value of the third integer is r.

Significance of an integer

The integer is described as the whole number which does not provide a fractional value. The integer also does not have a fractional component.

Determination of the uniqueness of the integers

Let the equation of the integer a can be expressed as:

$a=dQ+R$

Here, a is the value of the first integer and d is the value of the positive integer.

Here, $Q\ne q,0\le R<d$and$R\ne r$ then the above equation can be expressed as:

$$\begin{gathered} dq+r=dQ+R \\ d(Q-q)=r-R \end{gathered}$$

It has been assumed that$r>R$ without the loss of the generality. Moreover,$0<r-R<d$ and hence$r-R$ is not divisible by d.

Now, if the above equation gets equal to zero, then$Q=q$ and$R=r$ becomes a contradiction. Hence,r andQ are described as distinct.

Thus, the integersr andQ are unique.