If the HCF of 657 and 963 is expressible in the form $657 x + 963 \times (- 15)$ , find x.

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answer is D.

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Detailed Solution

The given numbers are 657 and 963.
Euclid's Division Lemma states that, if two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition:

a = bq + r

where

0 \leq r \leq b

Calculation of HCF using Euclid’s division algorithm is as follows:

963 = 657 \times 1 + 306

\Rightarrow 657 = 306 \times 2 + 45

\Rightarrow 306 = 45 \times 6 + 36

\Rightarrow 45 = 36 \times 1 + 9

\Rightarrow 36 = 9 \times 4 + 0

\Rightarrow HCF (657, 963) = 9

Calculate x.

HCF = 657 x + 963 \times (- 15)

9 = 657 x + 963 \times (- 15)

\Rightarrow 9 = 657 x - 14445

\Rightarrow 657 x = 14445 + 9

\Rightarrow 657 x = 14454

\Rightarrow x = \frac{14454}{657}

\Rightarrow x = 22

Therefore, if the HCF of 657 and 963 is expressible in the form

657 x + 963 \times (- 15)

then the value of x is 22.
Hence, option 4 is correct.

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