Show that, for any positive integers a,b the set of all ma+nb (m,n positive integers) includes all multiples of (a,b) larger than ab.
This problem is from a section on the integers in Ch 1 of Birkhoff and MacLane which covers:
(a,b), gcd of a,b
[a,b], lcm of a,b
Division algorithm, a=bq+r, 0<=r<b
p prime: p|ab → p|a or p|b
(c,a)=1 and c|ab → c|b
(a,c)=1, a|m, and c|m → ac|m
There are s and t such that (a,b)=sa+tb, and for all s and t, sa+tb is a multiple of (a,b).
An answer within the context of the section would be appreciated. My apologies for posting this stickler: I violated one of my cardinal rules: NEVER, NEVER, look back at the problems. Like Lot’s wife, it turned me into a pillar of salt and stopped me dead in my tracks.