If the HCF of 408 and 1032 is expressible in the form $1032m – 408 \times 5$, find $m$.

Given: HCF of 408 and 1032 is expressible in the form $1032m\ –\ 408\ \times\ 5$.

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

Solution:

To find the value of $m$ we have to calculate the HCF of 408 and 1032.

Using Euclid’s lemma to get: 

• $1032\ =\ 408\ \times\ 2\ +\ 216$

Now, consider the divisor 408 and the remainder 216, and apply the division lemma to get:

• $408\ =\ 216\ \times\ 1\ +\ 192$

Now, consider the divisor 216 and the remainder 192, and apply the division lemma to get:

• $216\ =\ 192\ \times\ 1\ +\ 24$

Now, consider the divisor 192 and the remainder 24, and apply the division lemma to get:

• $192\ =\ 24\ \times\ 8\ +\ 0$

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

Therefore the HCF of 408 and 1032 is the divisor at this stage, i.e., 24.

Given that HCF of 408 and 1032 is expressible in the form $1032m\ –\ 408\ \times\ 5$.So,

$24\ =\ 1032m\ –\ 408\ \times\ 5$

$24\ =\ 1032m\ –\ 2040$

$24\ +\ 2040\ =\ 1032m$

$2064\ =\ 1032m$

$m\ =\ \frac{2064}{1032}$

$\mathbf{m\ =\ 2}$

So, the value of $m$ is 2.

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Updated on: 10-Oct-2022

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