Prove that if a and b are odd then 2gcd(a,b)=gcd(a+b,a−b)”

13. Prove that for all integers b, if b is odd then b is odd
13. Prove that for all integers b, if b is odd then b is odd
(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every nonzero integers a, b, a, b are relatively prime if and only if a and −b are relatively prime.
(A) If d=gcd(a,b) and m=lcm(a,b), prove that dm=|ab|. (B) Show that lcm(a,b)=ab if and only if gcd(a,b)=1 (C) Prove that gcd(a,c)=gcd(b,c)=1 if and only if gcd(ab,c)=1 for integers a, b, and c. (Abstract Algebra)
If a and b are positive integers, then gcd (a,b) = sa + tb. Prove that either s or t is negative.
Write an alternative gcd algorithm based on the following observations (arrange so that a > b): a. gcd(a, b) = 2gcd(a/2, b/2) if a and b are both even. b. gcd(a, b) = gcd(a/2, b) if a is even and b is odd. c. gcd(a, b) = gcd(a, b/2) if a is odd and b is even. d. gcd(a, b) = gcd((a + b)/2, (a ? b)/2) if a and b are both odd
correction ---> gcd(a,b) = lcm(a,b)
( Let a and be positive integers. Prove that god(a,b) = lama,b) if and only if a
Prove that if a,b,c,d e Z and aſc, b|c, and the GCD of a and b is d then ab|cd 8 Format BI U
22. Prove: if a, b, and c are odd, and a | b - c and a bc, then a | b and a c
22. Prove: if a, b, and c are odd, and a | b - c and a bc, then a | b and a c
Prove all non-zero integers a and b, if gcd(a, b) = d then for all non-zero integers x if a|x and b|x then ab|dx.
22. Prove: if a, b, and c are odd, and a | b - c and a bc, then a | b and a c