You need to apply the Euclidean Algorithm:

117 = 65x1 + 52.

65 = 52x1 + 13

52 = 13x4 + 0

Therefore 13 is the HCF (65, 117).

Now work backwards:

13 = 65 + 52x(-1)

= 65 + [117 + 65x(-1)]x(-1)

= 65x(2) + 117x(-1).

Therefore m = 2 and n = -1.

You should note however that m and n are NOT unique ....