jmc

Doubling the given equation tells us $$31\cdot 74=2294.$$Specifically $$31\cdot 74\equiv 1(\mathrm{mod}2293)$$and 74 is the multiplicative inverse of 31 modulo 2293.

If we triple the congruence we just found we get $$31\cdot 222\equiv 3(\mathrm{mod}2293).$$Therefore $n=\boxed{222}$.