Hi, I am having a bit of difficulty with two problems dealing with number theory. Down below is what I was able to figure out so far. Any help would be greatly appreciated for either problem. Thanks.
#1: Let a be a positive integer. What is the greatest common divisor of a and a+2?
Work: We now (a, a+2) divides both a and a+2, written as (a, a+2) | a and (a, a+2) | a+2. By divisibility properties, we know this means (a, a+2)m = a and (a, a+2)n = a+2 for some integers m and n.
I know that I should show that (a, a+2) divides any linear combination of a and a+2. I know that my answer will probably end up with two possibilites, which I believe is 1 and 2. However, I can't see how to prove it and how to proceed from the work I already have written above.
#2: Show that if a, b, and c are integers such that (a, b) = 1 and c | (a+b), then (c, a) = (a, b) (a, c).
Work: From the assumption, we know ma + nb = 1 and cp = (a,b) for some integers m, n, p. I also solved for b= cp-a and then plugged that into the other equation to get ma+n(cp-a)=1, but I don't see yet how that helps.
I know that somehow I need to use the fact that if a linear combination of two numbers equals 1, then the greatest common divisor of the two numbers is 1 to solve the problem. Any help would be greatly appreciated. Thank you so much.
#1: Let a be a positive integer. What is the greatest common divisor of a and a+2?
Work: We now (a, a+2) divides both a and a+2, written as (a, a+2) | a and (a, a+2) | a+2. By divisibility properties, we know this means (a, a+2)m = a and (a, a+2)n = a+2 for some integers m and n.
I know that I should show that (a, a+2) divides any linear combination of a and a+2. I know that my answer will probably end up with two possibilites, which I believe is 1 and 2. However, I can't see how to prove it and how to proceed from the work I already have written above.
#2: Show that if a, b, and c are integers such that (a, b) = 1 and c | (a+b), then (c, a) = (a, b) (a, c).
Work: From the assumption, we know ma + nb = 1 and cp = (a,b) for some integers m, n, p. I also solved for b= cp-a and then plugged that into the other equation to get ma+n(cp-a)=1, but I don't see yet how that helps.
I know that somehow I need to use the fact that if a linear combination of two numbers equals 1, then the greatest common divisor of the two numbers is 1 to solve the problem. Any help would be greatly appreciated. Thank you so much.