How do I do the following?:If a|(b+c) and (b,c) = 1, prove that (a,b)=1=(a,c)

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Robert · Accepted Answer

1
Assume that gcd(a, b) 6= 1. Let gcd(a, b) = k. Then k | a and k | b ⇒ a = ka1, b = kb1. Now since a | (b+ c) we
have b+ c = ma ⇒ kb1 + c = m · (ka1) ⇒ c = mka1 − kb1 ⇒ k | c contradicting the fact that gcd(b, c) = 1.
Similary if gcd(a, c) = k,

How do I do the following?: If a|(b+c) and (b,c) = 1, prove that (a,b)=1=(a,c)

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