Question

Find HCF of 81 and 237 and Express it as a linear combination of 81 and 237 that is HCF of 81, 237 equals 81x + 237y for some X and Y

Answer 1

there the value of x is 13 and y is -38

Answer 2

According to the Question

Since, 237 > 81

On applying Euclid’s division algorithm, we get

237 = 81 × 2 + 75 ...(i)

81 = 75 × 1 + 6 ...(ii)

75 = 6 × 12 + 3 ...(iii)

6 = 3 × 2 + 0 ...(iv)

Hence, and HCF (81, 237) = 3.  

In order to write 3 in the form of 81x + 237y,  we move backwards :

3 = 75 - 6 × 12     [From (iii)]

= 75 - (81 - 75 × 1) × 12   [Replace 6 from (ii)]

= 75 - (81 × 12 - 75 × 1 × 12)

= 75 - 81 × 12 + 75 × 12

= 75 + 75 × 12 - 81 × 12

= 75 ( 1 + 12) - 81 × 12

= 75 × 13 - 81 × 12  

= 13(237 - 81 × 2) - 81 × 12   [Replace 75 from (i)]

= 13 × 237 - 81 × 2 × 13 - 81 × 12

= 237 × 13 - 81 (26 + 12)

= 237 × 13 - 81 × 38

= 81 × (- 38) + 237 × (13)  

= 81x + 237y

Hence, x = - 38 and y = 13