Question

find the HCF of 96 and 404 Express it as a linear combination of X and Y

Answer 1

HCF of 96 and 404 is therefore 4 and it can be expressed as linear combination as 4 = 404 × (-1) + 96 × 5.

• Using the Euclidean technique, we can determine the HCF (highest common factor) of 96 and 404 as follows:

• To obtain the quotient and remainder, first divide 404 by 96:

404 = 96 × 4 + 20

• We then divide 96 by 20 to obtain another quotient and remainder as follows:

96 = 20 × 4 + 16

• This step is repeated until the remaining value is zero:

20 = 16 × 1 + 4

16 = 4 × 4 + 0

• We come to an end here because the final residue is 0. The final non-zero remainder, which is 4, is the HCF of the numbers 96 and 404.
• Working backwards from the aforementioned divisions, we can use Euclid's division lemma to express 4 as a linear combination of x and y.

• starting with the division that is next to last:

4 = 20 - 16 × 1

• 20 from the prior division is substituted as follows:

4 = (404 - 96 × 4) - 16 × 1

• Simplifying:

4 = 404 × (-1) + 96 × 5

• The HCF of 96 and 404 is therefore 4, and it may be represented as a linear combination of x and y as follows:

4 = 404 × (-1) + 96 × 5

∴The HCF of 96 and 404 is therefore 4 and it can be expressed as linear combination as 4 = 404 × (-1) + 96 × 5.

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Answer 2

EXPLANATION.

Find the HCF of 96 and 404 express it as a linear combination of x and y.

First we try to understand the definitions of HCF.

HCF :

• Full form of HCF is "Highest common factor".
• It is the largest common factor of all the given numbers.

We can write 96 as,

⇒ 96 = 2 x 2 x 2 x 2 x 2 x 3.

⇒ 96 = 2² x 2² x 2 x 3.

⇒ 96 = 4 x 4 x 6.

The highest common factor of 96 is 4.

We can write 404 as,

⇒ 404 = 2 x 2 x 101.

⇒ 404 = 2² x 101.

⇒ 404 = 4 x 101.

The highest common factor of 404 is 4.

∴ The HCF of 96 and 404 is 4.

Now, we need to express it as a linear combination of X and Y.

With the help of Euclid division lemma we express it as a linear combination.

For any two positive integers,

⇒ a = bq + r.

⇒ 0 ≤ r < b.

We can write it as,

Dividend = Divisor x Quotient + Remainder.

96  and  404.

⇒ 404 = 96 x 4 + 20. - - - - - (1).

96 and 20.

⇒ 96 = 20 x 4 + 16. - - - - - (2).

20 and 16.

⇒ 20 = 16 x 1 + 4. - - - - - (3).

16 and 4.

⇒ 16 = 4 x 4 + 0. - - - - - (4).

Now, we again see that HCF is 4.

For linear combinations,

⇒ 4 = 20 - 16 x 1. - - - - - (from equation 3, we get).

⇒ 4 = 20 - (96 - 20 x 4) x 1. - - - - - (from equation 2, we get).

⇒ 4 = 20 - (96 x 1) + (20 x 4) x 1.

⇒ 4 = 20 - (96 x 1) + (20 x 4) x 1.

⇒ 4 = (404 - 96 x 4) + (20 x 4) x 1.

⇒ 4 = 404 x (-1) + 96 x 5.

∴ The linear combination can be express by  404 x (-1) + 96 x 5.