Express the hcf of 234 and 111 as 234x and111y. where x and y are integers

(- 9 × 234 ) + (19 × 111 )

Step-by-step 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

a .4y4

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