For all positive integers \(n\), the \(n\)th triangular number \(T_n\) is defined as \(T_n = 1+2+3+ \cdots + n\). What is the greatest possible value of the greatest common divisor of \(4T_n\) and \(n-1\)?
Not sure if I undestand your question!
Example: The 100th triangular number is: 5050. So, T(n) =T(100) ==5050. Then you say "What is the greatest possible value of the greatest common divisor of 4T(n) and n - 1? You want the GCD(4*5050 and 100 - 1) =GCD(20,200 , 99) ??? This does NOT make any sense to me, due to "4T(n)". Because of the presence of 4 in "4T(n)", the greatest value of the GCD will always be a 4 !!