Congruence properties of indices of triangular numbers multiple of other triangular numbers

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple other numbers. We analyze the congruence properties indices \(\xi\) Remainders in relations modulo \(k\) come always pairs whose sum equal \((k-1)\), include 0 and only \((k-1)\) if prime, or odd power a even square plus one minus two. If twice number \(n\), set remainders includes also \(n\) \((n^{2}-1)\) has factors, multiples factor following certain rules. Algebraic expressions are found for function its with several exceptions. This approach eliminates those values not providing solutions.