Corollary 2.3.6

Proof. Suppose \(a\) and \(b\) are not relatively prime, that is, they have a common factor \(d>1\). \(d\) divides the left hand side of \(ra+sb=1\), so it also divides the right hand side, a contradiction. QED

Corollary 2.3.7

Proof. Let \(p_1^{e_1}\cdots p_m^{e_m}\) be a factorization of \(a\). Let \(q_1^{e_1}\cdots q_n^{e_n}\) be a factorization of \(b\). Then \(p\in\{p_1,\ldots,p_m,q_1,\ldots,q_n\}\). So it’s either in \(\{p_1,\ldots,p_m\}\) or in \(\{q_1,\ldots,q_n\}\). QED

A question about the book’s proof of Corollary 2.3.7: Why does \(p\) divide \(rab\)? Because it divides \(ab\).

Proposition 2.3.8

Proof. (a) \(m\) is in the intersection of \({\mathbb Z}a\) and \({\mathbb Z}b\), so it’s divisible by both.

(b) If \(n\) is divisible by both \(a\) and \(b\), it’s in both \({\mathbb Z}a\) and \({\mathbb Z}b\). Hence it’s in their intersection \({\mathbb Z}m\). QED

Corollary 2.3.9

Proof. Look at their prime factorizations. QED