If $d$ is the $H C F$ of 45 and 27, then $x, y$ satisfying $d = 27 x + 45 y$ are :

x = 2, y = 1

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x = 2, y = - 1

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x = - 1, y = 2

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x = - 1, y = - 2

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Solution

The correct option is A $x = 2, y = - 1$
Applying Euclid's division lemma to $27$ and $45$ , we get
$45 = 27 \times 1 + 18$ ...(1)
$27 = 18 \times 1 + 9$ ...(2)
$18 = 9 \times 2 + 0$ ...(3)
Since the remainder is zero, therefore, last divisor $9$ is the $H C F$ of $27$ and $45$ .
From (2), we get
$9 = 27 - 18 \times 1$
$= 27 - (45 - 27 \times 1) \times 1 [u s i n g (1)]$
$= 27 - 45 \times 1 + 27 \times 1 \times 1$
$= 27 - 45 + 27$
$= 54 - 45$
$\Rightarrow 9 = 27 \times 2 - 45 \times 1$ ...(4)

Comparing (4) with $d = 27 x + 45 y$ , we get
$d = 9, x = 2 a n d y = - 1$

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