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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