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Answer 1

heya..


HCF of 126 and 35 :- 


Prime Factors of 35 = 5 × 7 

Prime factors of 126 = 2 × 3 × 3 × 7 

∴ common factors = 7 

∴ HCF = common factors = 7 


Now, A/C to question, 

HCF = 126A + 35B = 7 

⇒18 × 7A + 5 × 7B = 7 

⇒ 18A + 5B = 1 , here many solutions possible because given one equation and two variables.

Let A = 2 and B = -7 then, 18 × 2 - 5 × 7 = 1 

So, A = 2 and B = -7 is A solution of this equation .


Now, LHS = A.B/H 

PUT A = 2 , B = -7 and H = 7 

Then, A.B/H = 2 × -7/7 = -2 = RHS

Hence , proved//



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Answer 2

                           ===================

Let a = 126 and b = 35.

Using Euclid's Division Lemma,

First step:

⇒ 126 = 35 × 3 + 21

Second step :

⇒ 35 = 21 × 1 + 14

Third step :

⇒ 21 = 14 × 1 + 7

Forth step :

⇒ 14 = 7 × 2 + 0

So, H.C.F is 7.

Therefore H=7.

Now,

⇒ 7 = ( 21 - 14 × 1 )   Third step

⇒ 7 = { 21 - ( 35 - 21 × 1 ) × 1 }  Second step

⇒ 7 = { 21 - 35 × 1 + 21 × 1 }

⇒ 7 = ( 21 - 35 + 21 × 1 )

⇒ 7 = ( 21 + 21 × 1 - 35 )

⇒ 7 = { 21 ( 1 + 1 ) - 35 }

⇒ 7 = ( 21 × 2 - 35 )

⇒ 7 = [ ( 126 - 35 × 3 )2 - 35 ]   First step

⇒ 7 = [ 126 × 2 - 35 × 3 × 2 - 35 ]

⇒ 7 = [ 126 × 2 - 35 × 6 - 35 ]

⇒ 7 = [ 126 × 2 - 35 ( 6 + 1 ) ]

⇒ 7 = [ 126 × 2 - 35 × 7 ]

Here 7 is H.C.F ( H ) so,

⇒ H = 126 × 2 - 35 × 7

And it is given that H = 126 × A + 35 × B.

∴  126 × 2 - 35 × 7 = 126 × A + 35 × B

By comparing coefficients :

A = 2 and B = -7.

Now,

⇒ ( A × B ) ÷ H = -2

By substituting the values of H,A and B.

⇒ ( 2 × -7 ) ÷ 7 = -2

⇒ ( -14 ) ÷ 7 = -2

∴  -2 = -2.

Proved .

Hope it helps !