Express the HCF of 468 and 222 as 468x + 222y where x, y
Question:

Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.

Solution:

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.

Given integers are 468 and 222 , where $468>222$

By applying Euclid’s division lemma, we get $468=222 \times 2+24$.

Since the remainder $\neq 0$, so apply division lemma on divisor 222 and remainder 24

$222=24 \times 9+6$

Since the remainder $\neq 0$, so apply division lemma on divisor 24 and remainder 6

$24=6 \times 4+0$

We observe that remainder is 0. So the last divisor 6 is the H.C.F. of 468 and 222 from we have

$6=222-24 \times 9$

$\Rightarrow \quad 6=222-(468-222 \times 2) \times 9$      [Substituting $24=468-222 \times 2$ ]

$\Rightarrow \quad 6=222-468 \times 9+222 \times 18$

$\Rightarrow \quad 6=222 \times 19-468 \times 9$

$\Rightarrow \quad 6=222 y+468 x$, where $x=-9$ and $y=19$.