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Let $n$ be an integer greater than $3$. Find a formula for $\gcd(n, n + 3)$ for each of the cases :

$1)$ $n \equiv 0\mod 3$
$2)$ $n \equiv 1\mod 3$
$3)$ $n \equiv 2\mod 3$

Any help would be greatly appreciated.

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What have you tried? –  heropup Apr 21 at 9:23
I don't quite understand what is going on, ive calculated n such that 6+3k ≡ 0(mod 3), 7+3k ≡ 1(mod 3) and 8+3k ≡ 2(mod 3) for any k in N. –  user144626 Apr 21 at 9:27
There will be different cases. –  Satvik Mashkaria Apr 21 at 9:27
You are thinking way too hard about it. Why don't you start with some small cases and see what pattern develops? Try computing $\gcd(1,4), \gcd(2,5), \gcd(3,6), \gcd(4,7), \ldots$, etc. What do you see? How can you prove the pattern? –  heropup Apr 21 at 9:29
so gcd(3,n) is either 3 or 1 –  user144626 Apr 21 at 9:32

2 Answers 2

Apply the Euclidean algorithm. $\gcd(n, n+3) = \gcd(3, n)$. What's $\gcd(3, n)$ in each of the three cases?

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Suppose $d|n$ and $d|(n+3)$. Then $d|3 = (n+3)-n$. Thus $d=1,3$.

So the gcd of $n$ and $n+3$ is either $1$ or $3$.

Can you see what it must be for each case?

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For questions tagged homework, I recommend against simply doing the entire question for the person asking, especially when they have not shown evidence of their own personal effort to solve the question. –  heropup Apr 21 at 9:34
No problem, it is fixed –  fretty Apr 21 at 9:35

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