Show that a^2~0 or a^2~1 in Mod 4

In summary, using the division algorithm, we can show that for all integers, a^2 is equivalent to either 0 or 1 in mod 4, with even numbers being equivalent to 0 and odd numbers being equivalent to 1. This is accomplished by squaring the formula for even numbers, a = 2q + 0, and for odd numbers, a = 2q + 1. Therefore, a^2~0 or a^2~1 in mod 4 for all integers.
  • #1
PsychonautQQ
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Homework Statement


If a I san integer, show that a^2~0 or a^2~1 in mod 4 (~ represent equivalence)


Homework Equations





The Attempt at a Solution


my ATTEMPT:
I started with the division algorithm..
a = 2q + 1 for all odd numbers
a = 2q + 0 for all even numbers
then I squared the formula for even numbers...
a^2 = 4q^2 + 0... but q is arbitrary so
a^2 ~ 0 mod 4 for even numbers

then I did the same for odd numbers
a^2 = 4q^2 + 4q + 1
and then I said since q was arbitrary again I could say
a^2 = 4q + 1
a^2~1 mod 4 for odd numbers


is this the correct approach?
 
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  • #2
Yep, clean work.
 

1. What does it mean to show that a^2 is congruent to 0 or 1 in Mod 4?

Showing that a^2 is congruent to 0 or 1 in Mod 4 means that when a number a is divided by 4, the remainder is either 0 or 1. This is denoted by the notation a^2~0 or a^2~1.

2. What is Mod 4?

Mod 4, or "modulo 4", is a mathematical operation that calculates the remainder when a number is divided by 4. It is used to determine patterns and relationships between numbers in a given set.

3. How do you prove that a^2~0 or a^2~1 in Mod 4?

To prove that a^2~0 or a^2~1 in Mod 4, you can use the definition of congruence in modular arithmetic. This states that a and b are congruent modulo n if and only if a-b is divisible by n. Therefore, to prove a^2~0 or a^2~1 in Mod 4, you need to show that a^2-0 or a^2-1 is divisible by 4.

4. Can you provide an example of a number that satisfies a^2~0 or a^2~1 in Mod 4?

One example of a number that satisfies a^2~0 or a^2~1 in Mod 4 is 6. When 6 is divided by 4, the remainder is 2. However, when 6^2 is divided by 4, the remainder is 0. This satisfies the condition of a^2~0 or a^2~1 in Mod 4.

5. What implications does showing a^2~0 or a^2~1 in Mod 4 have?

Showing that a^2~0 or a^2~1 in Mod 4 can have various implications in different fields of mathematics. It can be used to prove certain number properties, such as divisibility, and can also be applied in number theory and cryptography. It can also help in determining patterns and relationships between numbers in a given set.

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