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Express the HCF of 468 and 222 as $468x + 222y$ where x, y are integers in two different ways.
Given: 468 and 222.
To do: Here we have to express the HCF of the given pair of integers as $468x\ +\ 222y$.
Solution:
To express the HCF of the given pair of integers as $468x\ +\ 222y$ we need to calculate the HCF. So,
Using Euclid's division algorithm to find HCF:
Using Euclid’s lemma to get:
- $468\ =\ 222\ \times\ 2\ +\ 24$ ...(i)
Now, consider the divisor 222 and the remainder 24, and apply the division lemma to get:
- $222\ =\ 24\ \times\ 9\ +\ 6$ ...(ii)
Now, consider the divisor 24 and the remainder 6, and apply the division lemma to get:
- $24\ =\ 6\ \times\ 4\ +\ 0$ ...(iii)
The remainder has become zero, and we cannot proceed any further.
Therefore the HCF of 468 and 222 is the divisor at this stage, i.e., 6.
Expressing the HCF of 468 and 222 as $468x\ +\ 222y$:
$6\ =\ 222\ –\ 24\ \times\ 9$ {from equation (ii)}
$6\ =\ 222\ –\ [468\ –\ 222\ \times\ 2]\ \times\ 9$ {from equation (i)}
$6\ =\ 222\ –\ 468\ \times\ 9 +\ 222\ \times\ 18$
$6\ =\ 222\ \times\ 19\ –\ 468\ \times\ 9$
$6\ =\ 468(-9)\ +\ 222(19)$
So,
$6\ =\ 468x\ +\ 222y$, where $\mathbf{x\ =\ -9\ \ and\ \ y\ =\ 19}$
Updated on: 10-Oct-2022
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