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Q37E

Expert-verifiedFound in: Page 344

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.

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*.

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

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:

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

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* and *Q* are described as distinct.

Thus, the integers *r* and *Q* are unique.

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