Answer:
(- 9 × 234 ) + (19 × 111 )
Explanation:
Using the division algorithm to find the hcf
If a and b are any positive integers, then there exists unique positive integers q and r such that
a = bq + r → 0 ≤ r ≤ b
If r = 0 then b is a divisor of a
Repeated use of the algorithm allows b to be found
here a = 234 and b = 111
234 = 2 × 111 + 12 → (1)
111 = 9 × 12 + 3 → (2)
12 = 4 × 3 + 0 ← r = 0
Hence hcf = 3
We can now express the hcf (d) as
d = ax + by where x, y are integers
From (2)
3 = 1 × 111 - 9 × 12
From (1)
3 = 1 × 111 - 9( 1 × 234 - 2 × 111)
= 1 × 111 - 9 × 234 + 18 × 111
= - 9 × 234 + 19 × 111 ← in required form
with x = - 9 and y = 19