Writing and Proof: (c) Is the following proposition true or false? Justify your conclusion.For each a e Z, if a # 0 (mod 3), then a² = 1 (mod 3).

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Description: Writing and Proof: (c) Is the following proposition true or false? Justify your conclusion.For each a e Z, if a # 0 (mod 3), then a² = 1 (mod 3).

The given proposition is as follows. For each a∈ℤ, if a≢0mod 3, then a2≡1mod 3. The given…

Answered: Is the following proposition true or…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Writing and Proof:

(c) Is the following proposition true or false? Justify your conclusion.
For each a e Z, if a # 0 (mod 3), then a² = 1 (mod 3).

Expert Solution

Step 1

The given proposition is as follows.

For each $a \in ℤ$ , if $a ≢ 0 (m o d 3)$ , then $a^{2} \equiv 1 (m o d 3)$ .

The given proposition is true.

This can be proved as shown below.

Consider $a \in ℤ$ with $a ≢ 0 (m o d 3)$ .

Note that, $a ≢ 0 (m o d 3)$ implies that a leaves a non zero remainder when divided by 3.

The possible remainders are 1 and 2.

Thus, if $a ≢ 0 (m o d 3)$ , then the remainder when a is divided by 3 is either 1 or 2.

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