Question

Use euclid s algorithm,to find the hcf of 65 and 117 and then express it in the form of 65x+117y , where x and y are integers

Answer 1

In order to find H.C.F of 65 and 117 we have to do following steps :-

⇒ 117 = 65 × 1 + 52 __(1)

⇒ 65 = 52 × 1 + 13 __(2)

⇒ 52 = 13 × 4 + 0

∴ H.C.F of 65 and 117 = 13

From eq (2) we get,

⇒ 65 = 52 × 1 + 13

⇒ 65 - (52 × 1) = 13

⇒ 65 - 52 = 13

From eq(1) we get,

⇒ 65 - (117 - 65) =  13

⇒ 65 + 117( - 1 ) + 65 = 13

⇒ 65(2) + 117(-1) = 13

Hence, H.C.F is in form of 65x + 117y where x = 2 and y = - 1.

Answer 2

Answer:

x = 2 and y = -1

Step-by-step explanation:

To find the hcf of 65 and 117, we will use Euclid's Division

a = bq +r

117 = 65 × 1 + 52 ----> [ 2 ]

65 = 52 x 1 + 13 -----> [1]

52 = 13 x 4 + 0

HCF  of 65 and 117 = 13 = 13

13 = 65m + 117n

From equation [ 1] we get ,

13 = 65 - 52 x 1

From equation [2] we get ,

52 = 117 - 65 x 1

Than backwords ,

13 = 65 + 52x(-1)

13 = 65 + [117 + 65x(-1)]x(-1)

13 = 65x(2) + 117x(-1).

x = 2 and y = -1

Hence, H.C.F of 65x + 117y, where x = 2 and y = - 1.