Number Theory Questions

• Start date
• #1
pupeye11
100
0

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 $$\cong$$0 (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)

• #2
Staff Emeritus
Homework Helper
22,178
3,314
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 $$\cong$$0 (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...

• #3
pupeye11
100
0
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?

• #4
Staff Emeritus
Homework Helper
22,178
3,314
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??

• #5
pupeye11
100
0
That is the problem, I haven't been able to find one in the book or in my notes.

• #6
Staff Emeritus
Homework Helper
22,178
3,314
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...

• #7
pupeye11
100
0
that ax gives us a number equal to the equivalence class [0], i.e. divisible by 78

• #8
Staff Emeritus
Homework Helper
22,178
3,314
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??

• #9
pupeye11
100
0
I am not really sure what you are getting at.

• #10
pupeye11
100
0
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?

• Last Post
Replies
7
Views
441
• Last Post
Replies
12
Views
401
• Last Post
Replies
7
Views
417
• Last Post
Replies
2
Views
356
• Last Post
Replies
2
Views
358
• Last Post
Replies
4
Views
264
• Last Post
Replies
9
Views
413
• Last Post
Replies
5
Views
341
• Last Post
Replies
3
Views
437
• Last Post
Replies
2
Views
448
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.632300853729248, "perplexity": 582.5820397893729}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710829.5/warc/CC-MAIN-20221201153700-20221201183700-00807.warc.gz"}