If the HCF of 657 and 963 is expressible in the form $657x + 963 \times (–15)$, find $x$.

Given: HCF of 657 and 963 is expressible in the form  $657x\ +\ 963\ \times\ (–15)$.

To find: Here we have to find the value of $x$.

Solution:

To find the value of $x$ we have to calculate the HCF of 657 and 963.

Using Euclid’s lemma to get: 

• $963\ =\ 657\ \times\ 1\ +\ 306$

Now, consider the divisor 657 and the remainder 306, and apply the division lemma to get:

• $657\ =\ 306\ \times\ 2\ +\ 45$

Now, consider the divisor 306 and the remainder 45, and apply the division lemma to get:

• $306\ =\ 45\ \times\ 6\ +\ 36$

Now, consider the divisor 45 and the remainder 36, and apply the division lemma to get:

• $45\ =\ 36\ \times\ 1\ +\ 9$

Now, consider the divisor 36 and the remainder 9, and apply the division lemma to get:

• $36\ =\ 9\ \times\ 4\ +\ 0$

The remainder has become zero, and we cannot proceed any further. 

Therefore the HCF of 657 and 963 is the divisor at this stage, i.e., 9.

Given that HCF of 657 and 963 is expressible in the form $657x\ +\ 963\ \times\ (–15)$. So,

$9\ =\ 657x\ +\ 963\ \times\ (–15)$

$9\ =\ 657x\ -\ 14445$

$9\ +\ 14445\ =\ 657x$

$14454\ =\ 657x$

$x\ =\ \frac{14454}{657}$

$\mathbf{x\ =\ 22}$

So, the value of $x$ is 22.

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Updated on: 10-Oct-2022

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