# Greatest common divisor of two numbers $a$ and $b$

So, in the image, we have the definition of the GCD. The only part of the text I don't understand is when it says that the set of common divisors is bounded above by the largest of $a$, $b$, $-a$, $-b$. I don't understand the reference to "$-a$, $-b$" ? I thought that the two numbers $a$ and $b$ are always positive, so what gives ? Thank you for the help !

• The greatest common divisor is also defined for negative numbers. For example, $gcd(-20,15)=5$ – ThePortakal May 10 '15 at 23:51
• Ok, but if I had gcd(-9,-3). the set of common divisors is -3,-1,1,3. How does what the text says apply ? – user108343 May 11 '15 at 0:01
• So, if you have two numbers $-9$ and $-3$, a common divisor cannot be bigger than $\max \{ a , b , -a , -b \} =\max \{ -9 , -3 , 9 , 3 \} = 9$ – ThePortakal May 11 '15 at 0:04
• Ahhhhhhhhh okkkkkk. I don't know why, but the way it was written confused me ! I feel ashamed of myself for this. By the way, you should answer if you want me to accept the answer (I think) Thank you again. – user108343 May 11 '15 at 0:10
• I don't think you have anything to be ashamed of. Absolutely everyone has these kinds of confusions. You recognized your difficulty and took steps to resolve it. Your question was clear and complete. I think you should be proud of how well you handled this. – MJD May 11 '15 at 0:16

If you have two numbers $a=-9$ and $b=-3$, a common divisor cannot be bigger than $\max \{a,b,-a,-b \} = \max \{-9,-3,9,3 \} = 9$.
So, all common divisors are $\leq 9$ which leads to the greatest common divisor exists and it is unique.