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Question

Fond the HCF of 65 and 117 and express it in the form 65n + 117n.


Solution

Euclid's Division formula :-
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 equals 65 minus left square bracket 117 minus 65 cross times 1 right square bracket rightwards double arrow 13 equals 65 minus left square bracket 117 minus 65 right square bracket rightwards double arrow 13 equals 65 minus 117 plus 65 rightwards double arrow 13 equals 65 plus 65 minus 117 rightwards double arrow 13 equals 2 cross times 65 minus 117
​​​​​​​
= 65 x 2 + 117 x [-1 ]

m = 2
n = -1

Therfore . Therefore H.C.F. of 65 and 117 is of the form 65m + 117n, where m = 2 and n = –1.

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