# Expression for finding common elements in two series

#### RTn

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 kâˆˆN. 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.?

#### idontknow

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.