I can only use induction or the well-ordering principle as aproof method. Prove:

Question

I can only use induction or the well-ordering principle as aproof method. Prove: "Suppose that a and b are positiveintegers. If a+b is prime, prove that gcd(a,b) = 1. Please provide some details on what this proof will looklike. I am so confused! Thank you so much. I can only use induction or the well-ordering principle as aproof method. Prove: "Suppose that a and b are positiveintegers. If a+b is prime, prove that gcd(a,b) = 1. Please provide some details on what this proof will looklike. I am so confused! Thank you so much.

Explanation / Answer

I can only use induction or the well-ordering principle as aproof method. Prove: "Suppose that a and b are positiveintegers. If a+b is prime, prove that gcd(a,b) = 1. Please provide some details on what this proof will looklike. I am so confused! Thank you so much.
A AND B ARE POSITIVE INTEGERS
A+B IS PRIME
TPT GCD[A,B]=1
LET US ASSUME IT IS NOT CORRECT
THAT IS LET GCD[A,B]=D WHERE D IS ANY POSITIVE INTEGER GREATER THAN1
THEN
D|A AND D|B
HENCE D|(A+B)
BUT IT IS GIVEN THAT A+B IS PRIME.
SO OUR ASSUMPTION THAT GCD[A,B]=D >1 IS NOT CORRECT
HENCE GCD[A,B]=1

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