I'm looking specifically at a question and answer in the student solutions for the 10th edition of Gallian's Abstract Algebra.
Q. Prove that every integer that is a common multiple of every member of a finite set of integers is a multiple of the least common multiple of those integers.
A. Let $a$ be the least common multiple of every element of the set and $b$ be any common multiple of every element of the set. Write $b = aq + r$ where $0 ≤ r ≤ a$.
My issue is that last '$\le$' where I expected '$<$'. Am I missing a subtlety here? Don't we always assume that $r < a$ when we use the division algorithm? When at the end of the proof we show that $r \ge a$ and derive a contradiction, it doesn't seem so much like a contradiction as a proof that $r=a$, based on this setup.
