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Find the HCF of 81 and 237 and express it as a linear combination of 81 and 237.

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3, 94 \times 237 + (- 275) \times 81

3, 94 \times 237 + (+ 275) \times 81

3, 95 \times 238 + (- 275) \times 81

3, 94 \times 237 + (- 277) \times 82

answer is A.

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Detailed Solution

Given Integer = 81 ,237 Here,

81 < 237

Now, applying division lemma,

237 = 81 \times 2 + 75

…………….(1)
So we can see the remainder

75 \neq 0

Again from division lemma,

81 = 75 \times 1 + 6

…………………(2)
We can see the new remainder

6 \neq 0

And the new divisor= 75,

75 = 6 \times 12 + 3

…………………(3)
When we take new divisor = 6 So, new remainder =3

6 = 3 \times 2 + 0

Now we can see the remainder at this stage is 0.
Here, for the representation of HCF as the linear combination of the provided numbers, take equation (3) as,

3 = 75 - 6 \times 12

\Rightarrow 3 = 75 - (81 - 75 \times 1) \times 12

\Rightarrow 3 = 75 - 12 \times 81 + 12 \times 75

\Rightarrow 3 = 13 \times 75 - 12 \times 81

\Rightarrow 3 = 13 \times (237 - 81 \times 2) - 12 \times 81

\Rightarrow 3 = 13 \times 237 - 26 \times 81 - 12 \times 81

\Rightarrow 3 = 13 \times 237 - 38 \times 81

Suppose, x=12 and y=-38 so we get the linear equation as,

\Rightarrow 3 = 237 x + 81 y

\Rightarrow 3 = 13 \times 237 - 38 \times 81 + 237 \times 81 - 237 \times 81

\Rightarrow 3 = (13 \times 237 + 237 \times 81) + (- 38 \times 81 - 237 \times 81)

\Rightarrow 3 = (13 + 81) \times 237 + (- 38 - 237) \times 81

\Rightarrow 3 = 94 \times 237 - 275 \times 81

\Rightarrow 3 = 94 \times 237 + (- 275) \times 81

Here, 3 is the HCF of given number which can be expressed as

94 \times 237 + (- 275) \times 81

.
Correct option is 1.

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