  It is known that, for any positive non-square integer multiplier $k$, there
is an infinity of multiples of triangular numbers which are triangular numbers.
We analyze the congruence properties of the indices $\xi$ of triangular numbers
that are multiples of other triangular numbers. We show that the remainders in
the congruence relations of $\xi$ modulo k come always in pairs whose sum
always equal $\left(k-1\right)$, always include 0 and $\left(k-1\right)$, and
only 0 and $\left(k-1\right)$ 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 $\left(n^{2}-1\right)$ and if $k$ has integer factors,
the set of remainders include multiples of a factor following certain rules.
Finally, algebraic expressions are found for remainders in function of $k$ and
its factors. Several exceptions are noticed and superseding rules exist between
various rules and expressions of remainders. This approach allows to eliminate
in numerical searches those $\left(k-\upsilon\right)$ values of $\xi_{i}$ that
are known not to provide solutions, where $\upsilon$ is the even number of
remainders. The gain is typically in the order of $k/\upsilon$, with
$\upsilon\ll k$ for large values of $k$.
