Question

using Euclid algorithm to find HCF of 1190 and 1445 Express the HCF in the form and 1190m + 1445n

Answer 1

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Since 1445 > 1190 , we apply the division lemma to 1445 and 1190 to get ,
=> 1445 = 1190 × 1 + 255 ......... ( 1 )

since the remainder 255 ≠ 0 , we apply the division lemma to 1190 and 255 ,
we get .
=> 1190 = 255 × 4 + 170 .............. ( 2 )

since the remainder 170 ≠ 0 , we apply the division lemma to 255 and 170 , we get
=> 255 = 170 × 1 + 85 ................ ( 3 )

since the remainder 85 ≠ 0 , we apply the division lemma to 170 and 85 ,
we get ,
=> 170 = 85 × 2 + 0 ................. ( 4 )

The remainder has now become zero , so our procedure stops, since the Divisor at this step is 85 ,
THE HCF of 1190 and 1445 is 85.

$hcf \: (1445 \: and \: 1190) \: = 85 \\ \\ now \: . \\ 85 = 255 - 170 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: from \: (3) \\ \\ =[ 1445 - 1190 \times 1 ]- \: [1190 - 255 \times 4 ]\\ [from \: (1) \: and \: \: from \: (2)] \\ \\ = 1445 - 1190 - 1190 + 255 + 4 \\ \\ = 1445 - 2 \times 1190 + (1445 - 1190) \times 4 \\ [from \: (1) ]\\ \\ = 1445 - 2 \times 1190 + 4 \times 1445 - 4 \times 1190 \\ = 5 \times 1445 - 6 \times 1190 \\ = > 85 = 1445(5) + ( - 6)1190 \\ \\ we \: \: compare \: \: with \: - \\ 85 = 1190m \: + 1445n \: \\ given - \\ m \: = - 6 \\ n = 5$
so , the answer is .
$85 \: m \: = - 6 \: and \: \: n \: = 5$

Answer 2

First we have to find the HCF of 1190 and 1445

Since, 1445 is greater than 1190

So 1445 is the dividend and 1190 is the divisor.By the process of Euclid's division lemma.

1445=1190×1+255.....(1)

1190=255×4+170.......(2)

255=170×1+85...........(3)

170=85×2+0...............(4)

So, we can say that 85 is the HCF of of(1190,1445)

First see all the equation above and find in which equation 85 which is the HCF of (1190,1445) comes as a reminder, I.e 3rd equation

Now,

225=170×1+85

85=255-170×1

85=(1445-1190)-(1190-255×4).....from(1) and(2)

85=1445-1190-1190-(1445-1190)×4........from(1)

85=(1445-2×1190)-1445×4+1190×4

85=1445-2×1190+1445×4+1190×4

85=1445×5+1190(-6)

So,therefore m=-6 and=5