Given, we need to express the H.C.F. of 468 and 222 as 468 x + 222 y where x, y are integers in two different ways.
So, here the integers are: 468 and 222, and 468 > 222
Then, by applying Euclid’s division lemma, we get
468 = 222 x 2 + 24……… (1)
Since the remainder ≠ 0, so apply division lemma on divisor 222 and remainder 24
222 = 24 x 9 + 6………… (2)
Since the remainder ≠ 0, so apply division lemma on divisor 24 and remainder 6
24 = 6 x 4 + 0……………. (3)
We observe that remainder is 0.
So, the last divisor 6 is the H.C.F. of 468 and 222
Now, in order to express the HCF as a linear combination of 468 and 222, we perform
6 = 222 – 24 x 9 [from (2)]
= 222 – (468 – 222 x 2) x 9 [from (1)]
= 222 – 468 x 9 + 222 x 18
6 = 222 x 19 – 468 x 9
= 468(-9) + 222(19)
∴ 6 = 468 x + 222 y, where x = -9 and y = 19.
