# Document 10581888

advertisement
```c Dr Oksana Shatalov, Spring 2015
Math 220-Homework 9
Due Wednesday 11/11 at the beginning of class
PART A
Problems from the textbook:
• Section 5.3 # 1b, 6(b,c,d,e (in part (e) state, prove or disprove the converse of parts 3&amp;4 of Proposition 5.3.3 only))
PART B
1. Let a, b, c ∈ Z. Determine the truth or falsehood of the following statements.
(a)
gcd(a, 0) = a.
(b)
Let a and b be not both zero. Then gcd(a, b) = gcd(−a, b).
(c)
The set Z − Z+ of integers is closed with respect to multiplication.
(d)
0|b only if b = 0.
(e)
If a|c and b|c, then ab|c.
(f)
If a|b and b|a then a = b.
2. Prove by induction that for every positive integer n the following statements hold:
(a) 2 + 6 + 10 + . . . + (4n − 2) = 2n2 .
(b) n3 + 2n is divisible by 3. (Hint: (a + b)3 = a3 + b3 + 3a2 b + 3ab2 )
1
1
1
n
(c)
+
+ ... +
=
.
2&middot;3 3&middot;4
(n + 1)(n + 2)
2(n + 2)
(d) 3|(22n − 1).
(e) 5 is a factor of 7n − 2n .
3. (a) Use the Euclidean Algorithm to determine gcd(374, 946).
(b) Find integers x and y such that 374x + 946y = gcd(374, 946).
1
```
{"found_math": false, "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": 0, "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, "url": "https://studylib.net/doc/10581888/", "fetch_time": 1638344341000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2021-49/segments/1637964359093.97/warc/CC-MAIN-20211201052655-20211201082655-00091.warc.gz", "warc_record_offset": 598154031, "warc_record_length": 12396, "token_count": 428, "char_count": 1084, "score": 3.90625, "int_score": 4, "crawl": "CC-MAIN-2021-49", "snapshot_type": "longest", "language": "en", "language_score": 0.6700635552406311, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00000-of-00064.parquet"}