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Question

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

Typical Answer

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

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?

Jose Arnaldo Bebita Dris
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  • I learned about that Typical Answer via FredH's comment to this answer. – Jose Arnaldo Bebita Dris Mar 02 '19 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 02 '19 at 10:39
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    See https://math.stackexchange.com/questions/2322680/gcd-inequality-with-products-and-exponents and https://math.stackexchange.com/questions/1023288/need-proof-of-lcm-inequality and https://math.stackexchange.com/questions/1871278/number-theory-problem-fractions-and-gcd-please-help and Theorem 5.1 in https://books.google.com.au/books?id=3iHPwAS7F2AC&pg=PA53&lpg=PA53&dq=gcd+inequalities&source=bl&ots=v4u7UDw26T&sig=ACfU3U1Z66RYGIxbWYFn7eIC3U7McesIyQ&hl=en&sa=X&ved=2ahUKEwjT-_rRsOPgAhUJVH0KHX3uCOgQ6AEwCnoECAIQAQ#v=onepage&q=gcd%20inequalities&f=false – Gerry Myerson Mar 02 '19 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 02 '19 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 02 '19 at 15:30

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