  We use modular symmetric designs to study the existence of Hadamard matrices
modulo certain primes. We solve the $7$-modular and $11$-modular versions of
the Hadamard conjecture for all but a finite number of cases. In doing so, we
state a conjecture for a sufficient condition for the existence of a
$p$-modular Hadamard matrix for all but finitely many cases. When $2$ is a
primitive root of a prime $p$, we conditionally solve this conjecture and
therefore the $p$-modular version of the Hadamard conjecture for all but
finitely many cases when $p \equiv 3 \pmod{4}$, and prove a weaker result for
$p \equiv 1 \pmod{4}$. Finally, we look at constraints on the existence of
$m$-modular Hadamard matrices when the size of the matrix is small compared to
$m$.
