Basic Divisibility Theory

Definition 50   We say that a divides b, written a|b, if there is a natural number n with an=b. Note that a|b is a sentence, not a number.

With this definition we can provide proofs of the following:

Proposition 7.1   If a|b and b|c then a|c.

Proposition 7.2   If a|b and a|c then a|b+c.

Proposition 7.3   If a|b and b|a then a=b.

Proposition 7.4   If a|b and a|b+c then a|c.

Proposition 7.5   For any a, a|a.

Proposition 7.6   For any a, 1|a.

Proposition 7.7   For any a, a|0.

Proposition 7.8   If a|b then $a\leq b$.

Find counterexamples for the following and then conjecture what additional hypotheses might make the statement true:

1.

If a|bc then either a|b or a|c.

2.

Either a|b or b|a.

3.

If a|c and b|c then ab|c.

4.

If a|c and b|c then a+b|c.

Definition 51   A prime number p is one with the property that exactly two natural numbers divide p, namely 1 and p.

Proposition 7.9   Neither 0 nor 1 is prime.

Proposition 7.10   For any number a and any prime p either p|a or if n|p and n|a is n=1.