Express the hcf/lcm of 48 and 18 as a linear combination

18x + 48y = 6, where x = 3, y = - 1

Solution:

Given numbers are 48 and 18.

Write it using linear combination, a = bq + r, where 0 ≤ r < b

48 = 18 * 2 + 12

⇒ 18 = 12 * 1 + 6 (write 18 using the previous remainder 12)

⇒ 12 = 6 * 2 + 0 (write 12 using the previous remainder 6)

Remainder is zero. So, the process stops.

∴ HCF (18, 48) = 6

Now, Express 6 using 18 and 48.

6 = 18 - 12 * 1

6 = 18 - (48 - 18 * 2)

6 = 18 - 48 + 18 * 2

6 = 18 * 3 - 48 * 1

6 = 18 * 3 + 48 * (-1)

6 = 18x + 48y

18x + 48y = 6, where x = 3, y = - 1

Hence 18x + 48y = 6 as the linear combination.