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