Show that $\gcd(ca,cb) = |c|\gcd(a,b)$

Question

I understand this:

If $\gcd(a,b)=d$ and $\gcd(ac,bc)=e$, then $d|a \Rightarrow dc|ac$ and $d|b \Rightarrow dc|bc$, so this is equivalent to multiplying $c$ by $\gcd(a,b)$.

What I don't understand so well is why the $|c|$ (absolute value). Why can't $c$ be negative?

What is $\gcd(-20,-30)$? – Angina Seng Dec 31 '19 at 05:57 — Angina Seng, Dec 31 '19 at 05:57
@LordSharktheUnknown It's 10 – Segmentation fault Dec 31 '19 at 06:51 — Segmentation fault, Dec 31 '19 at 06:51
That's $10$ times $\gcd(2,3)$ then. – Angina Seng Dec 31 '19 at 12:57 — Angina Seng, Dec 31 '19 at 12:57

score 1 · Accepted Answer · answered Dec 31 '19 at 05:58

1

By definition, gcd of two integers is the largest positive integer that divides each of the integers. (See here)

answered Dec 31 '19 at 05:58

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Show that $\gcd(ca,cb) = |c|\gcd(a,b)$

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