AbstractWe 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 conjectural sufficient condition for the existence of ap‐modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a primep, we conditionally solve this conjecture and therefore thep‐modular version of the Hadamard conjecture for all but finitely many cases when, and prove a weaker result for. Finally, we look at constraints on the existence ofm‐modular Hadamard matrices when the size of the matrix is small compared tom.