Expression for finding common elements in two series

RTn

Apr 2019
1
0
Europe
Hello,
First of, I am sorry if I am posting it in a wrong section. If so, can someone please move this thread to the appropriate section?

Now, my problem:
I have a series of the form S1=x(x+1)
and another series S1/k, for any k∈N. Now I want to find the values where the elements of two series are equal. For example, let k be 3, then the intersection of the two series gives 2,30,420,5852,81510,1135290. Can a formula be derived to find these ways without resorting to equating the two series and deriving quadratic roots i.e. using series formulae or triangular numbers etc.?
 
Dec 2015
1,084
169
Earth
Without equating them then use divisibility rule.
\(\displaystyle k|x(x+1)\) or S1 is divisible by k.
S1(mod)k=0 .To continue use modular arithmetics.