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Found in: Page 344

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Let a be an integer and d be a positive integer. Show that the integers q and 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.

The integers r and Q are unique.

See the step by step solution

## Step 1: 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.

## Step 2: 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.

## Step 3: 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 and $R\ne r$ then the above equation can be expressed as:

$dq+r=dQ+R\phantom{\rule{0ex}{0ex}}d\left(Q-q\right)=r-R$

It has been assumed that $r>R$ without the loss of the generality. Moreover, $0 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 and Q are described as distinct.

Thus, the integers r and Q are unique.