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How can I prove that $a$ and $b$ are coprime iff they have no common prime divisors?

I had this question on an exam and I have no clue where to start.

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    that is a definition – 2017-02-02
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    How do you define "coprime"? – 2017-02-02
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    @Arnaldo oh well... – 2017-02-02
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    @lulu $a$ and $b$ are coprime means that gcd(a,b)=1 – 2017-02-02
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    Ok...but you can't have a common divisor without having a common prime divisor. – 2017-02-02
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    They have a common prime divisor $\!\iff\! $ they have a common divisor $d>1$ (any prime divisor of $d)\ \ $ – 2017-02-02
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    @BillDubuque that helped – 2017-02-02

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First $a,b$ coprime $\implies$ they have no common prime divisor.

Assume they had a common prime divisor. Let the common prime divisor be $p$. Then $\gcd(a,b)=1$ is divisible by $p$. Contradiction.

Second, they have no common prime divisor $\implies$ $a,b$ are coprime.

Assume they aren't coprime. Then they have a common divisor $d \neq 1$. Let a prime factor of $d$ be $p$. $p$ is a common prime factor of $a,b$. Contradiction.

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    I have deleted my answer, despite your answer coming after mine. In general I like to avoid duplicate answers from my side (although it wasn't my fault in this case). Thank you for pointing this out. – 2017-02-03
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    @астонвіллаолофмэллбэрг You didn't have to do that. But thanks. – 2017-02-03
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    @SCB You are welcome. Actually, after reading this meta post:http://meta.math.stackexchange.com/questions/25735/unfair-up-voting-of-high-reputation-not-having-a-special-answer?cb=1 I have decided to avoid writing duplicate answers from my side due to issues faced by the user here. Since I have control over my answer, I have deleted it. It's ok for me, questions come and go, right! – 2017-02-03
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    @астонвіллаолофмэллбэрг You do realize that I have more reputation than you. I should be the one who had deleted, if anyone was going to delete. – 2017-02-03
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    @SCB No, I'm just scared, for whatever reason, that the up votes may be skewed. Nothing to do with reputation or presentation. It has become a norm of mine, (what my mother says is the norm, and she told me to do this) to delete my own answers unless they are heavily up voted or duplicates appear much later, In our case, the answers appeared nearly at the same time and also were extremely similar, so to avoid double reading I deleted my own answer, for benefit of questioner. So it was never a question of reputation, but that meta post anyway shook me. – 2017-02-03