Question

let a,b,q and r be positive integers such that a>b , r<b , a=bq+r and hcf(a,b)=2K×hcf(b,r). write the value of k ​

Answer 1

Answer:

answer of your question

a=bq+r

Let a common factor of 'a' and 'b' be 'c'.

So a=cA and b=cB, where 'A' and 'B' are integers.

Substituting these values in the first equation, we get

cA = cBq + r

In the left hand side we have a multiple of 'c'. Hence, the right hand side should also be a multiple of 'c'.

cBq is a multiple of 'c'. So for 'cBq + r' to be a multiple of 'c', the second term 'r' must be a multiple of 'c'.

Hence, we can write 'r' as cR, where 'R' is some integer.

Hence, 'c' is a common factor of 'b' and 'r'. (Proved)

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