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Question

If 'd' is the HCF of 30,72,find the value of 'x' and 'y' satisfying d=30x+72y.
 


Solution

EUCLID'S DIVISION ALGORITHM:

Given positive integers a and b , there exist unique integers q and r satisfying

a = bq + r ,

0 ≤ r < b 

*************************************************************************************

Applying Euclid's division lemma to 30 and 72 

Since 72 > 30

72 = 30 × 2 + 12 ------------( 1 )

30 = 12 × 2 + 6   ------------( 2 )

12 = 6 × 2 + 0     -----------( 3 )

The remainder has now become zero,

Since the divisor at this stage is 6 , 

The HCF of 30 and 72 = 6

now from ( 2) 

30 = 12 × 2 + 6

Rearrange this 

6 = 30 - 12 × 2

⇒ 6 = 30 - [ (72 - 30 × 2 ) × 2 ]   { from ( 1 ) }

⇒ 6 = 30 - 72 × 2 + 4 × 30     [ using distributive property ]

⇒ 6 = 30 × ( 5 ) + 72 × ( -2 )  -------( 4 )

according to the problem ,

d = 30x + 72y --------------------------( 5 )

compare ( 4 ) and ( 5 ) ,

x = 5 and y = -2

Note : these x , y values are not unique.


 

Mathematics

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