Thread: division theorem

Let m be the least common multiple of a and b, and let c be a common multiple of a and b. Show that m divides c. Hint: use the division theorem on m and c , and show that the remainder r is a common multiple of a and b, hence r=0.

m is the LCM of a and b, so a|m, b|m
c is a CM of a and b, so a|c, b|c
m<=c, since m is the LCM.
so, use the division theorem we set up the equation c=mq+r. for some integers q and r, where 0<=r<m.
Rearrange the equation which becomes r=c-mq.
since a|c, a|m, b|c, b|m, we get a|r, b|r.
We know m is the LCM, so r should be 0, otherwise m will not be the LCM.
since r = 0, we get c=mq, which completes the proof m|c.