jmc

We can write the condition on $t$ as $$11\cdot t\equiv 36(\mathrm{mod}100).$$Then, multiplying both sides by $9$, we have $$99\cdot t\equiv 324\equiv 24(\mathrm{mod}100).$$The left side, $99t$, is congruent modulo $100$ to $-t$, so we have $$-t\equiv 24(\mathrm{mod}100)$$and therefore $$t\equiv -24(\mathrm{mod}100).$$The unique two-digit positive solution is $t=-24+100=\boxed{76}$. Indeed, we can check that $11\cdot 76=836$, which does end in $36$.