Question

Use Euclid' s algorithm to find HCF of 1190 and 1445.express the HCF in the form 119m + 1445n.

Answer 1

Hope this helps you ☺☺

Answer 2

aloha user !!

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we know that a = bq + r ( Euclid's division lemma )

( a is for the dividend, b is for the divisor, q is for the quotient and r for remainder )

and b must be smaller than r.

so we will find the HCF of 1190 and 1445.

1445 = 1190 × 1 + 255
1190 = 255 × 4 + 170
255 = 170 × 1 + 85
170 = 85 × 2 + 0

now we got the remainder 0.
the HCF is 85 ( the divisor )

now we need to express it in the form 119m + 1445n.

119m + 1445n = 85
                          
solving RHS: 

now we have to take two values of a which on subtraction would provide us with the number 85.

85 = 255 - 170

now 255 should be written which on subtraction would give 255. similarly we are going to write the values for 170.

the equations which we will use are:

1445 = 1190 × 1 + 255 => 1445 - 1190 = 255
1190 = 255 × 4 + 170 => 1190 - 255 × 4 = 170

85 = ( 1445 - 1190 ) - ( 1190 - 255 × 4 )
85 = 1445 - 1190 - 1190 + (255) × 4                
85 = 1445 -1190 × 2 + ( 1445 -1190 ) × 4   ----- ( ✯ )

what we did in  ( ✯ ):
[ ( - 1190 - 1190 => 2 × -1190 )] 

$also$

255 = 1445 -1190

now,

85 = 1445 -1190 × 2 + ( 1445 -1190 ) × 4 

85 = 1445 -1190 × 2 + 1445 × 4 - 1190 × 4

85= 1445 × 5 - 1190 × 6

85 = 1445 × 5 + 1190 × (-6)

85 = 1190 × (-6) + 1445 × 5 

clearly we can see that the value of m is  -6 and the value of n is 5.


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hope it helps !!

peace out !!