Question

find hcf of 65 ad 117 ad find a pait of integral valuevof m ad n such that hcf=65m+117n.

Answer 1

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Let a = 117 and b =65.

Using Euclid's Division Lemma,

First step :

117 = 65 × 1 + 52

Second step:

65 = 52 × 1 + 13

Third step :

52 = 13 × 3 + 0

Therefore,H.C.F = 13.

Now,

13 = 65 - ( 52 × 1 ) Second step

13 = 65 - [ ( 117 - 65 × 1 ) 1 ] First step

13 = 65 - 117 × 1 + 65 × 1

13 = 65 + 65 - 117 × 1

13 = 65 ( 1 + 1 ) - 117 × 1

13 = 65 × 2 - 117 × 1 ------ ( 1 )

Given,

H.C.F = 65m + 117n

13 = 65m + 117n ------- ( 2 )

From ( 1 ) and ( 2 ) we get,

65 × 2 - 117 × 1 = 65 m + 117n

By comparing coefficients,

m = 2 and n = -1.


Hope it helps !!

Answer 2

HCF(65,117) = 13

13 = 65 (2) + 117 (1)

m= 2, n=1