Find the HCF of and and express it as a linear combination of and .
Step 1 : Find the HCF of given numbers
Given integers are and such that ,
Applying division lemma to and , we get
…….(i)
Since the remainder is . So, apply the division lemma to the divisor and the remainder to get
……(ii)
We consider the new divisor and the new remainder and apply division lemma, to get
……(iii)
We consider the new divisor and the new remainder and apply division lemma, to get
……(iv)
At this stage the remainder is zero. So, that last divisor or the non-zero remainder at the earlier stage i.e. is the HCF of and .
Step 2 : Express the HCF in the form of a linear combination of the given numbers
Given numbers: and .
Rewriting the equation (iii)
By replacing in the above expression, from equation (ii)
By replacing from equation(i) as
, where and .
Hence, the HCF of and is and the linear combination can be expressed as , where and .