Question

find the HCF of 65 and 117 and Express it in the form of 65m + 117n​

Answer 1

Hcf= 13.

m= 1

n = 2

Hope it will help you...........

Answer 2

Since 117 > 65, by using Euclid's division lemma to 117 and 65 to get,



117 = 65 * 1 + 52 ------(i)

Since, r ≠ 0, therefore, by using Euclid's division lemma to 65 and 52 to get,

65 = 52 * 1 + 13 -------(ii)

Since, r ≠ 0, therefore, by using Euclid's division lemma to 52 and 13 to get,

52 = 13 * 4 + 0

Since, r = 0, therefore, the divisor of the last step will be the HCF of given two numbers.

Therefore, HCF(117, 65) = 13

As the HCF(117, 65) can be expressed in the form of 65m + 117n, so it must equal to 13 as it's the HCF(117, 65).

13 = 65m + 117n

From equation (i),

117 = 65 * 1 + 52

13 = 65 - 52 * 1

From equation (ii),

65 = 52 * 1 + 13

117 = 65 - 52 * 1 -------(iii)

Hence, by combing,

13 = 65 - 52

13 = 65 - (117 - 65 * 1) [(as 117 - 65 = - 52)]

As 65 is common,

13 = 65 * 2 (117 * 1)

13 = 65 * 2 + 117 * (-1)

Now, we can see that,

13 is expressed in the form of 65m + 117n.

Here, 2 can be replaced by m and - 1 replaced by n.

So, we get an expression as,

13 = 65m + 117n

Which is our required expression.

Therefore, 13 can be expressed in the form 65m + 117n.

Therefore, the values of m and n are 2 and - 1 respectively.

$\boxed{\bold{m = 2}}$

And

$\boxed{\bold{n = - 1}}$

$\boxed{\bold{Note:}}$

Never to write Euclid's division algorithm directly as algorithm is the Step-by-step process to solve the problem.

So, if you write all the steps correctly then your answer is correct.

Never to write Euclid's division algorithm directly.