Chapter 1: Q15E (page 49)

Determine necessary and sufficient conditions on $x and c$ so that the following holds: for any $a, b,$ if $ax \equiv bx \mod c$ , then $a \equiv bmodc$ .

Short Answer

Expert verified

The necessary and sufficient condition for $x and c$ is $(a - b)$ must be divisible by $c$ as the $GCD (c, x)$ is equal to $1$ .

Step by step solution

Introduction

Fermat’s Little Theorem states that for any prime number $x$ and integer $i$ , the number i.e. $i^{x} - i$ is the factor of $x$ .

Given condition

The given is,

For any $a, b, c$

If $ax = bx modc$

To prove: $a = b \mod c$

Condition we can define from the given statement

If $ax = bx modc$

Then,

$\frac{c}{(a - b) x}$

If $a = bmodc$ ,

$\frac{c}{(a - b)}$

Explanation

So, here $c$ divides $(a - b) x$ then, $c$ must divide $(a - b) .$

For the value of $x$ , the $GCD (c, x)$ must equal to $1$ that ensure that $(a - b) x$ is divisible by $c$ .

Thus, in this case $(a - b)$ must be divisible by $c$ as the $GCD (c, x)$ is equal to $1$ .

Therefore, the necessary and sufficient condition for $x and c$ is $(a - b)$ must be divisible by $c$ as the $GCD (c, x)$ is equal to $1$ .

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Determine necessary and sufficient conditions on $x and c$ so that the following holds: for any $a, b,$ if $ax \equiv bx \mod c$ , then $a \equiv bmodc$ .

Short Answer

Step by step solution

Introduction

Given condition

Condition we can define from the given statement

Explanation

One App. One Place for Learning.

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Determine necessary and sufficient conditions on xandc so that the following holds: for anya,b, if ax≡bxmodc, thena≡bmodc .

Short Answer

Step by step solution

Introduction

Given condition

Condition we can define from the given statement

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Computer Systems

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Determine necessary and sufficient conditions on $x and c$ so that the following holds: for any $a, b,$ if $ax \equiv bx \mod c$ , then $a \equiv bmodc$ .