Solving for Solutions in Z12: X^2=4

In summary, The solutions to the equation X^2=4 in Z12 are X=2, 4, 6, 8, 10 and -2, -4, -6, -8, -10. The odd numbers cannot be solutions because their squares will yield an odd number, which is not congruent to 4 mod 12. The solutions are the numbers that are congruent to 2,4,6,8,10 mod 12.
  • #1
mikee
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Homework Statement

Find all solutions in Z12 to the equation X^2=4



Homework Equations





The Attempt at a Solution

This is my attempt, since it is in Z12, i wrote X^2=4(modulo 12) and by this the answers are trivial, X= 2, 4,8,... and also -2,-4...
this just seems to easy so i was wondering if i was doing it wrong
 
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  • #2
That's correct. It can't be any of the odd numbers because the odds squared will yield an odd number, which will still be an odd number mod 12.

0^2 = 0 mod 12 and 6^2 = 36 = 0 mod 12, so your answer is right.

Although you have to be careful with the ..., that could be interpreted as an exponential series, what you're saying is the numbers that are congruent to 2,4,8 and 10 mod 12.
 

1. What is a congruence in number theory?

A congruence in number theory is a relationship between two integers where the remainder of both numbers when divided by a third number is the same. This can be expressed as a ≡ b (mod n), meaning that a and b have the same remainder when divided by n.

2. How is congruence used in mathematics?

Congruence is used in a variety of mathematical concepts, such as modular arithmetic, cryptography, and geometry. It is also used to solve equations and prove theorems in number theory.

3. What is the difference between congruence and equality?

Congruence and equality are both ways to compare two mathematical objects. However, congruence compares the relationship between two objects, while equality compares the objects themselves. In other words, congruence looks at the similarity of two objects, while equality looks at their exactness.

4. How do you solve a congruence equation?

To solve a congruence equation, you can use modular arithmetic to manipulate both sides of the equation until you isolate the variable. This can involve adding, subtracting, multiplying, or dividing both sides by the same number. Once you have solved for the variable, you can check your solution by plugging it back into the original equation.

5. What are some real-life applications of congruences and number theory?

Congruences and number theory have many real-life applications, such as in cryptography, where they are used to create secure codes and protect sensitive information. They are also used in computer science, engineering, and physics to solve problems and make calculations more efficient. In addition, congruences and number theory are used in the study of prime numbers, which has applications in fields like chemistry, biology, and economics.

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