If d is the H C F of 56 and 72 , find x , \textrm{ } y satisfying d = 56 x + 72 y . Also, show that x and y are not unique.

Foundation

Mathematics Foundation

HCF and LCM

Properties of Real Number

Question

d

is the

H C F

56

and

72,

find

x, y

satisfying

d = 56 x + 72 y .

Also, show that

x

and

y

are not unique.

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Applying Euclids division lemma to

56

and

72

, we get

72 = 56 \times 1 + 16

Since the remainder

16 \neq 0

. So, we consider the divisor

56

and remainder

16

and apply division lemma to get

56 = 16 \times 3 + 8

..........

(i i)

We consider the divisor

16

and the remainder

8

and apply division algorithm to get

16 = 8 \times 2 + 0

We observe that the remainder at this stage is zero. Therefore, last divisor

8

(or the remainder at the earlier stage) is the

H C F

56

and

72

From

(i i)

, we get

8 = 56 - 16 \times 3

\Rightarrow 8 = 56 - (72 - 56 \times 1) \times 3

\Rightarrow 8 = 56 - 3 \times 72 + 56 \times 3

\Rightarrow 8 = 56 \times 4 + (- 3) \times 72

∴ x = 4

and

y = - 3

Now,

8 = 56 \times 4 + (- 3) \times 72

\Rightarrow 8 = 56 \times 4 + (- 3) \times 72 - 56 \times 72 + 56 \times 72

\Rightarrow 8 = 56 \times 4 - 56 \times 72 + (- 3) \times 72 + 56 \times 72

\Rightarrow 8 = 56 \times (4 - 72) + {(- 3) + 56} \times 72

\Rightarrow 8 = 56 \times (- 68) + (53) \times 72

∴ x = - 68

and

y = 53

Hence

x

and

y

are not unique.