This an example in my notes...

Encipering transformation: f(x) = 9x + 3(mod 100)

Finding the deciphering transformation:

Alice now wishes to find a number t such that 9t = 1 mod 100. She finds that 9 x 89 = 1 mod 100. She finds by trial and error (until she improves her arithmetic!) that 9 x 89 = 1 mod 100.

I was just wondering is there another quicker way that alice can find this t, rather than trial and error?

Thanks in advance!