Question

Find the HCF of 65 and 117 and express it in the form of 65m + 117n

Answer 1

Hello there !

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

Hope this Helped You !
mark as brainliest

Answer 2

let the two positive integers as a and b

a=117

b=65

a<b

a/b

Euclid's Division algorithm as follows :

117=65×1+52(1)

65=52×1+13(2)

52=13×4+0(3)

therefore HCF=13

to express it in the linear combination of the given two numbers,we start from last one step and successively eliminate the previous remainders as follows:

from (2),we have

13=65-52×1

13=65-(117-65×1)

13=65-117+65×1

13=65×2+117×(-1)

13=65-117+65×1

13=65m+117n[where m=2,n=-1]

hope it helps