Show How p Equals 1 or 5 in Z_6 (mod 6)

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In summary, the conversation is discussing a mathematical question about prime numbers and their congruency in the set Z_6 (mod 6). The question states that if p is not equal to 2, then 3 is prime and the task is to show that p can only be equal to 1 or 5 in this set. The participants in the conversation are clarifying the question and discussing their interpretations and proofs.
  • #1
Pearce_09
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Hello,
I am just going to post question i have tried it yet.. only because i don't understand the question.. so if you could help me read the question that would be good.. and if you want to give a hint feel free..

If p does not equal 2, 3 is prime, show that p = 1 or p = 5 in [tex] Z_6 [/tex] (mod 6)

ok what i don't understand is.. is the question saying that p does not also equal 3.. or is it just telling me that 3 is a prime.. cause that obvious..
or is it saying p does not equal 3 but its prime.. I am not sure.. so i can't really go any further.. thank you
 
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  • #2
I think it would mean that p is a prime other than 2 or 3.
 
  • #3
this seems ok (this is not a proof by any means,) but for p=5,7,11,13,17,19,23,29,31 it works. so It seems that our interpetation of the hypothesis on p is correct.
 
  • #4
awsome..thats what i was thinking to.. thanks!
 
  • #5
The question just means "if p is a prime not equal to 2 or 3". Exactly what it says. It says nothing about what 2 or 3 are at all, primes or not.

If we were to remove the requirement that p not be 2 or 3 then the statement would read: suppose p is a prime, show p is congruent to 1 or 5 mod 6. And that would be false since there are two primes that are not congruent to 1 or 5 mod 6. However all primes except 2 and 3 are congruent to 1 or 5 mod 6. now let's prove it...

I don't see why this meant you couldn't go further. the question was specifically not about the primes 2 or 3.
 

FAQ: Show How p Equals 1 or 5 in Z_6 (mod 6)

What does "p Equals 1 or 5 in Z_6 (mod 6)" mean?

The notation "p Equals 1 or 5 in Z_6 (mod 6)" is referring to the set of integers modulo 6. This means that we are looking at numbers from 0 to 5, and when we add or multiply them, we take the remainder after dividing by 6.

What is the significance of p equaling 1 or 5 in Z_6 (mod 6)?

When p equals 1 or 5 in Z_6 (mod 6), it means that p can be represented as either 1 or 5 when divided by 6. In other words, p is congruent to 1 or 5 modulo 6.

How can we show that p equals 1 or 5 in Z_6 (mod 6)?

We can show this by using modular arithmetic. We can first show that p is congruent to 1 modulo 6 by dividing p by 6 and checking if the remainder is 1. Similarly, we can show that p is congruent to 5 modulo 6 by dividing p by 6 and checking if the remainder is 5.

Why is it important to understand p equals 1 or 5 in Z_6 (mod 6)?

Understanding modular arithmetic, including p equals 1 or 5 in Z_6 (mod 6), is important in many areas of math and science. It has applications in cryptography, computer science, number theory, and more.

What are some real-life examples of using p equals 1 or 5 in Z_6 (mod 6)?

One real-life example is in the world of music. The 12-tone equal temperament system, which is used to tune musical instruments, is based on the concept of modular arithmetic. In this system, the octave is divided into 12 equal intervals, with each interval being represented by a number from 0 to 11 (mod 12). This system allows musicians to easily transpose music to different keys by using modular arithmetic.

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