Find the HCF of the following pair of integers and express it as a linear combination of them:
592 and 252

Given: 592 and 252

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: 
  • $592\ =\ 252\ \times\ 2\ +\ 88$   ...(I)

Now, consider the divisor 252 and the remainder 88, and apply the division lemma to get:
  • $252\ =\ 88\ \times\ 2\ +\ 76$   ...(ii)

Now, consider the divisor 88 and the remainder 76, and apply the division lemma to get:

  • $88\ =\ 76\ \times\ 1\ +\ 12$   ...(iii)

Now, consider the divisor 76 and the remainder 12, and apply the division lemma to get:
  • $76\ =\ 12\ \times\ 6\ +\ 4$   ...(iv)

Now, consider the divisor 12 and the remainder 4, and apply the division lemma to get:
  • $12\ =\ 4\ \times\ 3\ +\ 0$   ...(v)

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

Therefore the HCF of 592 and 252 is the divisor at this stage, i.e., 4.


Expressing the HCF as a linear combination of 592 and 252:

$4\ =\ 76\ –\ 12\ \times\ 6$   {from equation (iv)}

$4\ =\ 76\ –\ [88\ –\ 76\ \times\ 1]\ \times\ 6$   {from equation (iii)}

$4\ =\ 76\ –\ 88\ \times\ 6 +\ 76\ \times\ 6$

$4\ =\ 76\ \times\ 7\ –\ 88\ \times\ 6$

$4\ =\ [252\ –\ 88\ \times\ 2]\ \times\ 7\ –\ 88\ \times\ 6$   {from equation (ii)}

$4\ =\ 252\ \times\ 7\ –\ 88\ \times\ 14\ –\ 88\ \times\ 6$

$4\ =\ 252\ \times\ 7\ –\ 88\ \times\ 20$

$4\ =\ 252\ \times\ 7\ –\ [592\ –\ 252\ \times\ 2]\ \times\ 20$   {from equation (i)}

$4\ =\ 252\ \times\ 7\ –\ 592\ \times\ 20\ +\ 252\ \times\ 40$

$\mathbf{4\ =\ 252\ \times\ 47\ –\ 592\ \times\ 20}$


So, HCF of 592 and 252 is 4 and it can be expressed as  $4\ =\ 252\ \times\ 47\ –\ 592\ \times\ 20$.

Updated on: 10-Oct-2022

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