Show that \gcd(a,b)=|a| \iff a | b?

Aleena Kaiser 2022-04-25 Answered
Show that gcd(a,b)=|a|ab?
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Answers (2)

gunithd5
Answered 2022-04-26 Author has 8 answers
Step 1
Assume that a, bN. I use the following definition of D=gcd(a,b):
1) D is a common divisor of a and b;
2) Every integer dN which is a common divisor of a and b divides D.
Proposition 1: If gcd(a,b)=a then ab
Proposition 2: If ab, then gcd(a,b)=a
Proof: Since aa and ab, then any integer d such that da and db satisfies also the condition d|a.
From a|a, a|b and da we conclude that gcd(a,b)=a
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ophelialee4xn
Answered 2022-04-27 Author has 14 answers
Step 1
It is true that the final assertion holds if s=0 and t=1, but it is not true that that as+bt=az=b can only hold if s=0 and t=1 is false. Take, for example, a=2 and b=4. Then (5)a+3(b)=b.
Remember that, by definition, gcd(x,y)x and gcd(x,y)y. That gives one implication.
For the converse, remember that d|x and d|y implies dgcd(x,y).
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