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?




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) \times gcd(b,c) = gcd(3,3) \times gcd(3,3) = 3 \times 3 = 9




\therefore gcd(ab,c) \neq gcd(a,b)\times gcd(b,c)


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f \colon A \rightarrow B and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...