Could somebody answer part 1 and 2 Thank you Could somebody

Could somebody answer part (1) and (2)?

Thank you

Could somebody answer part (1) and (2)? C. Properties of Relatively Prime Integers Prove the following, for all integers a, b, c, d, r, and s. (Theorem 3 will be helpful.) 1 If there are integers r and s such that ra + sb = 1, then a and b are relatively prime. 2 If gcd(a, c) = 1 and c | ab, then c | b. (Reason as in the proof of Euclid?s lemma.) 3 If a|d

Solution

1.
Proof by contradiction. a and b are not relatively prime.
Hence, a and b share a common factor, call it m.
For any integers n and p, a = mn and b = mp.
ar + bs = 1
(mn)r + (mp)s = 1
m(nr + ps) = 1
nr + ps = 1/m
Contradiction since the sum of nr and ps, both integers, should result in an integer value yet
1/m is not an integer.
Hence, a and b are relatively prime.

2.
Proof by contradiction. Suppose that b is not divided by c.
Then, ab is divided by c. Since b is not divided by c, a must be divided by c.
But a and c are relatively prime because gcd(a, c) = 1.
Thus, contradiction.
Hence, b must be divided by c.