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

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x = 2, y = 1

x = 2, y = - 1

x = - 1, y = 2

x = - 1, y = - 2

answer is B.

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Detailed Solution

Given,

d

is the HCF of 45 and 27 and

d = 27 x + 45 y

.
We need to find the values of x and y.
According to Euclid’s Division Lemma if we have two positive integers

a

and

b

, then there exist unique integers

q

and

r

which satisfy the condition

a = bq + r

where

0 \leq r < b

.
Here, larger number

a

is dividend, smaller number

b

is divisor,

q

is a quotient and

r

is the remainder.
Applying Euclid's division lemma to

27

and

45

, we get

45 = 27 \times 1 + 18

... (1)
Here,

a = 45

b = 27

q = 1

and

r = 18

27 = 18 \times 1 + 9

... (2)
Here,

a = 27

b = 18

q = 1

and

r = 9

18 = 9 \times 2 + 0

... (3)
Here,

a = 18

b = 9

q = 2

and

r = 0

.
Since the remainder is zero, therefore, last divisor

9

is the HCF of

27

and

45

.
From (2), we get

9 = 27 - 18 \times 1

\Rightarrow 9 = 27 - (45 - 27 \times 1) \times 1

\Rightarrow 9 = 27 - 45 \times 1 + 27 \times 1 \times 1

\Rightarrow 9 = 27 - 45 + 27

\Rightarrow 9 = 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, y = - 1

Therefore,

x = 2

and

y = - 1

.
Hence, option 2 is correct.

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