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Suppose $a, b, c$ are 3 consecutive integers, and $b$ is an even number. Is it true that the three numbers are pairwise prime?

I have tested a few cases, for example $9, 10, 11$, and it seems to be true. How can I prove/disprove that?

Thank you.

Gareth Ma
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Deepanshu Singh
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  • Not sure what you are asking. Sure, three consecutive numbers can be pairwise relatively prime. They can't be all be primes (at least one of them is even and the triple $1,2,3$ does not work). – lulu Apr 05 '20 at 12:56
  • What do you think about $3,4,5$ ? –  Apr 05 '20 at 12:56
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    I think you are asking this: If $a,b,c$ are three consecutive positive integers, and $b$ is even, are they always pairwise co-prime? And the answer is yes. – TonyK Apr 05 '20 at 12:58
  • thank you @TonyK – Deepanshu Singh Apr 05 '20 at 13:01
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    I'll still answer here. Notice that $a,c$ are odd and their difference is 2. $\gcd(a,c)=\gcd(a,a+2)=\gcd(a,2)=1$ by euclidean algorithm. The remaining have differences 1, so similarly they've $\gcd 1$, completing the proof – Gareth Ma Apr 05 '20 at 14:49
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    Also, why is this question closed? It's pretty clear to me to be honest, perhaps edit TonyK's clarification into the question statement. – Gareth Ma Apr 05 '20 at 14:50

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