Pascal's triangles modulo $n$ can be considered as binary
operations on the set $\NN$ of nonnegative integers. To
every binary operation $f$ on a set $M$ an equivalence
relation $R$ on $M^2$ can be associated in which $(x,y)$ is
equivalent with $(u,v)$ if and only if $f(x,y)=f(u,v)$. The
equivalence $R$ can be considered as a $4$-ary relation on
$M$, and we can try to reconstruct $f$ from $R$, more
precisely, to define elementarily $f$ in $\str{M;R}$. (Some
abstract information about $f$ is used by this.) This
problem will be solved for Pascal's triangles modulo $n$ for
$1 \le n \le 6$. The  answer is positive for $n=4$ and
$n=6$, negative for $n=1$ and $n$ prime (even greater
than~$6$). As a corollary we obtain that the operations $+$,
$\times$ are definable in the structure $\strN{\EqB_6}$,
where $\EqB_6=\set{(x,y,u,v)\in\NN^4\mid
\binom{x+y}x\equiv\binom{u+v}u\pmod 6}$; the integer $6$
cannot be replaced by any smaller positive integer. The
above mentioned problem will be solved (positively, resp.
negatively) also for addition and multiplication on the set
$\NN$, resp. $\ZZ$, and for some other operations.
