What are the values of m² + n² modulo 4 in number theory?

[pupeye11]

Homework Statement

1) What are the possible values of m^{2} + n^{2} modulo 4?

2) Let d_{1}(n) denote the last digit of n (the units digit)

a) What are the possible values of d_{1}(n^{2})?

b) If d_{1}(n^{2})=d_{1}(m^{2}), how are d_{1}(n) and d_{1}(m) related?

3) a) Find all possible additive orders for a (mod 78)

b) for each order d in part a), find a number n with order d.

The Attempt at a Solution

1) Would this just be looking for the equivalence classes. So in this case the equivalence classes would be [0], [1], [2], [3]?

2) I honestly have no idea what this question is even asking.

3) a) Well if a was a number given I could rewrite the problem as ax \cong0 (mod 78). Where do I go from here in this case though?

b) Maybe this part will make more sense after I figure out a)

 

[micromass]

1) Would this just be looking for the equivalence classes. So in this case the equivalence classes would be [0], [1], [2], [3]?

No. Of course [0], [1], [2], [3] are the only possible equivalence classes. But you don't know that they actually occur. For example: 2=1²+1², so you see that [2] occurs as an equivalence class. Now, try to see which of the three others occurs.

2) I honestly have no idea what this question is even asking.

Take an arbitrary element n. What are the possible last digits of n²? For example: 4²=16, so 6 is a possible last digit. Now find the other ones.

3) a) Well if a was a number given I could rewrite the problem as ax \cong0 (mod 78). Where do I go from here in this case though?

What does ax\cong 0 (mod 78) mean? Just apply it's definition...

 

[pupeye11]

So for question 1. [0] can occur because 2^{2}+2^{2}= 0 mod 4.
[1] can occur because 2^{2}+1^{2}= 1 mod 4.
Is [3] the only one that can not occur?

As for question 2a, I went through and squared all numbers from 1 to 20, the only options would only be 0,1,4,5,6,9.

2b. it seems that the difference between them is always a multiple of 2.

3a. I am not sure what the definition of it is?

 

[micromass]

So for question 1. [0] can occur because 2^{2}+2^{2}= 0 mod 4.
[1] can occur because 2^{2}+1^{2}= 1 mod 4.
Is [3] the only one that can not occur?

As for question 2a, I went through and squared all numbers from 1 to 20, the only options would only be 0,1,4,5,6,9.

2b. it seems that the difference between them is always a multiple of 2.

This is all correct. But you will probably have to prove all of this...

3a. I am not sure what the definition of it is?

What does your textbook give as definition??

 

[pupeye11]

That is the problem, I haven't been able to find one in the book or in my notes.

 

[micromass]

That is the problem, I haven't been able to find one in the book or in my notes.

What does it mean to you? Just give me what you think it means. It doesn't have to be a rigorous definition...

 

[pupeye11]

that ax gives us a number equal to the equivalence class [0], i.e. divisible by 78

 

[micromass]

that ax gives us a number equal to the equivalence class [0], i.e. divisible by 78

Indeed, so 78 divides ax. But, what does divisibility mean??

 

[pupeye11]

I am not really sure what you are getting at.

 

[pupeye11]

For 3 part a, additive order of a modulo n is defined to be the smallest positive integer m that satisfies the congruence equation m*a \cong 0 (mod n). So in this case it'd be better to write a modulo n as m*a \cong 0 (mod n). m would be our additive order which means since n=78 our m=78/a?