# Inequalities involving the greatest common divisor

Question

What are some inequalities involving the greatest common divisor (GCD) function?

$$\gcd(a,bc) \leq \gcd(a,b)\gcd(a,c)$$

My Attempt

I tried checking the Wikipedia page, but the properties listed there for the greatest common divisor (GCD) function are all in the form of equations.

Can you recommend an online resource/textbook/reference that lists properties for the GCD function in the form of inequalities?

• I learned about that Typical Answer via FredH's comment to this answer. – Jose Arnaldo Bebita-Dris Mar 2 at 10:33
• I just found this Wolfram link, which lists two inequalities involving the GCD, LCM and the product of the arguments. – Jose Arnaldo Bebita-Dris Mar 2 at 10:39
• – Gerry Myerson Mar 2 at 11:48
• Thanks Gerry for providing those links! Do you happen to know the source for the Typical Answer mentioned in my post? – Jose Arnaldo Bebita-Dris Mar 2 at 11:58
• The example is true by LSH divides RHS $= (a\color{#c00}{(a,b,c)},bc),\,$ i.s. $\,(a,bc)\mid a\color{#c00}d,bc\, \Rightarrow\, (a,bc)\mid (a\color{#c00}d,bc)$ $\qquad$ – Bill Dubuque Mar 2 at 15:30