If HCF (65,117) is expressed in the form 65m+117n, then find the value of m.
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 ....