Wednesday, November 20, 2019

elementary number theory - How to prove or disprove that gcd(ab,c)=gcd(a,b)timesgcd(b,c)?



I'm new to proofs, and am trying to solve this problem from William J. Gilbert's "An Introduction To Mathematical Thinking: Algebra and Number Systems". Specifically, this is from Problem Set 2 Question 74. It asks:




How to prove or disprove that gcd(ab,c)=gcd(a,b)×gcd(b,c)?




What I've tried is to use the proposition that gcd(a,b)=ax+by to rewrite the whole equality, but I can't manage to equate the two statements.




Any help would be appreciated.


Answer



Notice if a=b=c=3, then



gcd(ab,c)=gcd(9,3)=3



while



gcd(a,b)×gcd(b,c)=gcd(3,3)×gcd(3,3)=3×3=9




gcd(ab,c)gcd(a,b)×gcd(b,c)


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