understanding proof of theorem(1) of greatest common divisor in 'higher algebra' by bernard and child?

Question

extract from book 'higher algebra' by bernard and child

i understood the 2 sentences in proof(see image in link above).

since a = bq + r, every common divisor of b and r is a divisor of a ; and, since r = a - bq, every common divisor of a and b is a divisor of r.

but i don't see how they prove the theorem.

can you help me understand this proof from authors point of view?

Answer 1

Just write down, what it means for a number $d$ to be a divisor:

• $d | b,r \Rightarrow b=kd,\; r=ld \Rightarrow a = bq+r = kdq+ld =d(kq+l) \Rightarrow d|a$
• similarly the other case

Edit after comment:

The author uses the following fact:

• Given two sets $A,B$. Then $A= B \Leftrightarrow A\subset B \mbox{ and } B\subset A$

Now apply this fact to the two sets:

• $A = \{d \in \mathbb{Z}| d|b \mbox{ and } b|r\}$
• $B = \{d \in \mathbb{Z}| d|b \mbox{ and } d|a\}$