Proving Modular Relations: b ≡ 1 (mod 2) ⇒ b² ≡ 1 (mod 8)

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In summary, a modular relationship, also known as modular arithmetic, involves finding the remainder of a division operation using the modulo symbol. To prove such a relationship, two numbers must have the same remainder when divided by a certain number. This concept has practical applications in various fields, such as computer science and engineering. Some common properties of modular arithmetic include commutative, associative, and distributive properties. However, it has limitations such as only working with whole numbers and difficulty in finding the inverse of a number in a modular system.
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John112
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[itex]\forall[/itex] b[itex]\in[/itex] [itex]Z[/itex] b [itex]\equiv[/itex] 1 (mod 2) [itex]\Rightarrow[/itex] b[itex]^{2}[/itex] [itex]\equiv[/itex] 1 (mod 8)

How do I go about proving this? Can the Chinese Remainder Theorem be used to prove this or is there something easier?
 
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Directly from the definition, [tex]b\conguent 1 (mod 2)[/tex] means that b= 1+ 2k[/tex] for some integer k. Then [itex]b^2= 4k^2+ 4k+ 1= 4(k^2+ k)+ 1[/itex]

Now consider two cases:
1) k is even: k= 2n for some integer n. What is [itex]4(k^2+ k)+ 1[/itex] in this case?

2) k is odd: k= 2n+ 1 for some integer n. What is [itex]4(k^2+ k)+ 1[/itex] in this case?
 

FAQ: Proving Modular Relations: b ≡ 1 (mod 2) ⇒ b² ≡ 1 (mod 8)

1. What is a modular relationship?

A modular relationship, also known as modular arithmetic, is a mathematical concept that involves finding the remainder of a division operation. This is often represented using the modulo symbol (mod) and is used to study patterns in numbers.

2. How do you prove a modular relationship?

To prove a modular relationship, you must show that two numbers have the same remainder when divided by a certain number, known as the modulus. This can be done through various methods, such as using the division algorithm or using congruence statements.

3. Can modular relationships be applied to real-life situations?

Yes, modular relationships have many practical applications in fields such as computer science, cryptography, and engineering. They can be used to solve problems involving repeating patterns, time zones, and error correction.

4. What are some common properties of modular arithmetic?

Some common properties of modular arithmetic include the commutative, associative, and distributive properties. Additionally, the addition, subtraction, and multiplication of two congruent numbers will result in another congruent number.

5. What are the limitations of modular relationships?

One limitation of modular relationships is that they only work with whole numbers. Additionally, division by zero is undefined in modular arithmetic. Another limitation is that it can be difficult to find the inverse of a number in a modular system.

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