Question

if the hcf of 65 and 117 is expressible in the form 65m-117n find the value of mn

Answer 1

Euclid's Division Lemma :-
a = bq +r

117 > 65

117 = 65 × 1 + 52 ----> [ 2 ]

65 = 52 x 1 + 13 -----> [1]

52 = 13 x 4 + 0

HCF = 13

13 = 65m + 117n

From [ 1] ,
13 = 65 - 52 x 1

From [2] ,
52 = 117 - 65 x 1 ----> [3]
Hence ,

13 = 65 - [ 117 - 65 x 1 ] ------> from [3]

= 65 x 2 - 117

= 65 x 2 + 117 x [-1 ]

m = 2
n = -1