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number theory

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For all positive integers n, the nth triangular number T_n is defined as $T_n = 1+2+3+...+ n$. What is the greatest possible value of the greatest common divisor of 4T_n and n - 2?

Apr 7, 2021

#1
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$T_n=\frac{n(n-1)}{2}$ and $4T_n=2n(n-1)=2n^2-2n$

And $\gcd(n-2, 2n^2-2n)=\gcd(n-2, 2n^2-2n-2n(n-2))$ by Euclidean algorithm.

And $2n^2-2n-2n(n-2)=2n^2-2n-[2n^2-4n]=2n$, so it remains to maximize $\gcd(n-2, 2n)=\gcd(n-2, 2n-2(n-2))=\gcd(n-2, 4)$ which has maximum value of $\boxed{4}$.

Apr 7, 2021