A question about inverse modulo a number.

If xy $equiv$ 1 (mod a), Can i write x $equiv$ $frac{1}{y}$ (mod a).

I was thinking about this while solving IMO 2005 P4, so can i write 2$^{p-2}$ + 3$^{p-2}$ + 6$^{p-2}$ $equiv$ $frac{1}{2}$ + $frac{1}{3}$ + $frac{1}{6}$ (mod p)