Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

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

The integers r and Q are unique.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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:

$a = d Q + R$

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

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

$d q + r = d Q + R d (Q - q) = r - R$

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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Discrete Mathematics and its Applications

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Short Answer

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

Step by Step Solution

Step 1: Identification of the given data

Step 2: Significance of an integer

Step 3: Determination of the uniqueness of the integers

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Discrete Mathematics and its Applications

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Short Answer

Let a be an integer and d be a positive integer. Show that the integers q and r with a=dq+r and 0≤r<d which were shown to exist in Example 5, are unique.

Step by Step Solution

Step 1: Identification of the given data

Step 2: Significance of an integer

Step 3: Determination of the uniqueness of the integers

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