Question

Express hcf of 468 ,222 as 468x+222y where x,y are integers in two different ways?

Answer 1

By Euclid’s division algorithm,

HCF of 468 and 222 is
468 = (222 x 2) + 24  ----------------------(1)
222 = (24 x 9) + 6   ------------------------(2)
24 = (6 x 4) + 0  

So the HCF of 468 and 222 is 6.
Now we have to write 6 as 468x + 222y

6 = 222 - (24 x 9)  --------------- [ from (2) ]

Now write 24 as (468 – 222 x 2) -------------- [ from (1) ]
⇒ 6 = 222 - {(468 – 222 x 2) x 9               
       = 222 - {468 x 9 – 222 x 2 x 9}
       = 222 - (468 x 9) + (222 x 18)
       = 222 + (222 x 18) - (468 x 9)
       = 222[1 + 18] – 468 x 9
       = 222 x 19 – 468 x 9
       = 468 x -9 + 222 x 19

So HCF of 468 and 222 is (468 x -9 + 222 x 19) in the form 468x + 222y.

Answer 2

Given: 468x + 222y

To show: HCF of 468 and 222 as 468x + 222y in two different ways.

Explanation:

HCF of 468 and 222 is found by division method:

Therefore, HCF (468,222) = 6

Now, we need to express the HCF of 468 and 222 as 468x + 222y where x and y are any two integers.

Now, HCF i.e. 6 can be written as,

HCF = 222 - 216 = 222 - (24 × 9)

Writing 468 = 222 × 2 + 24, we get,

⇒ HCF = 222 - {(468 – 222 x 2) × 9}

⇒ HCF = 222 - {(468 ×9) – (222 × 2 × 9)}

⇒ HCF = 222 - (468 × 9) + (222 × 18)

⇒ HCF = 222 + (222 × 18) - (468 × 9)

Taking 222 common from the first two terms, we get,

⇒ HCF = 222[1 + 18] – 468 × 9

⇒ HCF = 222 × 19 – 468 × 9

⇒ HCF = 468 × (-9) + 222 × (19)

Let, say, x = -9 and y =19

Then, HCF = 468 ×(x) + 222 ×(y)

Therefore, the HCF of 468 and 222 is written in the form of 468x + 222y where, -9 and 19 are the two integers.