Question

detail method of solving hcf of 65 and 117 and writing it in the form of 65m+117n

Answer 1

Write the bigger one in terms of the other, with quotient and reminder, when one number is divided by the other.

HCF(117, 65)
Write 117 = 65 * 1 + 52.        Now find HCF (65, 52)
Replace bigger number by the reminder. 
Continue like this, until one of them is 0.

HCF(117, 65)
 =  HCF(52, 65)            as 117 = 65 * 1 +  52
 = HCF(52, 13)            as 65 = 52 * 1 + 13
 = HCF(0, 13)              as 52  = 13 * 4 + 0      reminder 0.

So 13 is the HCF.
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13 = 65 - 52 = 65 - [ 117 - 65 ] = 2 * 65 - 117

13 = 65 * 2 - 117 * 1
m = 2  n = 1