Find the HCF of the following pair of integers and express it as a linear combination of them: 1288 and 575

Academic Mathematics NCERT Class 10

Given: 1288 and 575.

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:

$1288\ =\ 575\ \times\ 2\ +\ 138$ ...(i)

Now, consider the divisor 575 and the remainder 138, and apply the division lemma to get:

$575\ =\ 138\ \times\ 4\ +\ 23$ ...(ii)

Now, consider the divisor 138 and the remainder 23, and apply the division lemma to get:

$138\ =\ 23\ \times\ 6\ +\ 0$ ...(iii)

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

Therefore the HCF of 1288 and 575 is the divisor at this stage, i.e., 23.

Expressing the HCF as a linear combination of 963 and 657:

$23\ =\ 575\ –\ 138\ \times\ 4$ {from equation (ii)}

$23\ =\ 575\ –\ [1288\ –\ 575\ \times\ 2]\ \times\ 4$ {from equation (i)}

$23\ =\ 575\ –\ 1288\ \times\ 4\ +\ 575\ \times\ 8$

$\mathbf{23\ =\ 575\ \times\ 9\ –\ 1288\ \times\ 4}$

So, HCF of 1288 and 575 is 23 and it can be expressed as $23\ =\ 575\ \times\ 9\ –\ 1288\ \times\ 4$.

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Updated on: 10-Oct-2022

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