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Find the HCF of the following pair of integers and express it as a linear combination of them:
963 and 657
Given: 963 and 657
To do: Here we have to find the HCF of the given pair of integers and express it as a linear combination.
Solution:
Using Euclid's division algorithm to find HCF:
Using Euclid’s lemma to get:
- $963\ =\ 657\ \times\ 1\ +\ 306$ ...(i)
Now, consider the divisor 657 and the remainder 306, and apply the division lemma to get:
- $657\ =\ 306\ \times\ 2\ +\ 45$ ...(ii)
Now, consider the divisor 306 and the remainder 45, and apply the division lemma to get:
- $306\ =\ 45\ \times\ 6\ +\ 36$ ...(iii)
Now, consider the divisor 45 and the remainder 36, and apply the division lemma to get:
- $45\ =\ 36\ \times\ 1\ +\ 9$ ...(iv)
Now, consider the divisor 36 and the remainder 9, and apply the division lemma to get:
- $36\ =\ 9\ \times\ 4\ +\ 0$ ...(v)
The remainder has become zero, and we cannot proceed any further.
Therefore the HCF of 963 and 657 is the divisor at this stage, i.e., 9.
Expressing the HCF as a linear combination of 963 and 657:
$9\ =\ 45\ –\ 36\ \times\ 1$ {from equation (iv)}
$9\ =\ 45\ –\ [306\ –\ 45\ \times\ 6]\ \times\ 1$ {from equation (iii)}
$9\ =\ 45\ –\ 306\ +\ 45\ \times\ 6$
$9\ =\ 45\ \times\ 7\ –\ 306$
$9\ =\ [657\ –\ 306\ \times\ 2]\ \times\ 7\ –\ 306$ {from equation (ii)}
$9\ =\ 657\ \times\ 7\ –\ 306\ \times\ 14\ –\ 306$
$9\ =\ 657\ \times\ 7\ –\ 306\ \times\ 15$
$9\ =\ 657\ \times\ 7\ –\ [963\ –\ 657\ \times\ 1]\ \times\ 15$ {from equation (i)}
$9\ =\ 657\ \times\ 7\ –\ 963\ \times\ 15\ +\ 657\ \times\ 15$
$\mathbf{9\ =\ 657\ \times\ 22\ –\ 963\ \times\ 15}$
So, HCF of 963 and 657 is 9 and it can be expressed as $9\ =\ 657\ \times\ 22\ –\ 963\ \times\ 15$.