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 $9\ =\ 657\ \times\ 22\ –\ 963\ \times\ 15$.

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Updated on: 10-Oct-2022

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