Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.
HCF of 468 and 222 is found by division mehtod:
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.