Divisibility and Greatest Common Divisors

Definition 2.1. When a and b are integers, we say a divides b if b = ak for some k ∈ Z. We then write a | b (read as “a divides b”). Example 2.2. We have 2 | 6 (because 6 = 2 · 3), 4 | (−12), and 5 | 0. We have ±1 | b for every b ∈ Z. However, 6 does not divide 2 and 0 does not divide 5. Divisibility is a relation, much like inequalities. In particular, the relation 2 | 6 is not the number 3, even though 6 = 2 ·3. Such an error would be similar to the mistake of confusing the relation 5 < 9 with the number 9− 5. Notice divisibility is not symmetric: if a | b, it is usually not true that b | a, so you should not confuse the roles of a and b in this relation: 4 | 20 but 20 4. Remark 2.3. Learn the definition of a | b as given in Definition 2.1, and not in the form “ b a is an integer.” It essentially amounts to the same thing (exception: 0 | 0 but 0 0 is not defined), however thinking about divisibility in terms of ratios will screw up your understanding of divisibility in other settings in algebra. That is why it is best to regard Definition 2.1, which makes no reference to fractions, as the correct definition of divisibility.