Question

Find the HCF of $81$ and $237$ and express it as a linear combination of $81$ and $237$. What does inear combination mean?

Answer 1

In mathematics, a linear combinations of an expression constructed from a set of terms by multiplying each term by a constant and adding the results.
Example: a linear combination of $x$ and $y$ would be any expression of the form $ax+by$, where $a$ and $b$ are constant terms.
Between $81$ and $237$; $237$ is greater than $81$

From Division lemma of $237$ and $81$:

Step 1: $237=81\times 2+75$

Step 2: Since remainder $75\ne 0$,
division lemma is applied to $81$ and $75$ to get $81=75\times 1+6$

Step 3: Since remainder $6\ne 0$,
division lemma is applied to $75$ and $6$ to get $75=6\times 12+3$

Step 4: Since remainder $3\ne 0$,
division lemma is applied to $6$ and $3$ to get $6=3\times 2+0$

The remainder is zero in step 4.

Therefore, the divisor i.e.$3$ in this step is the H.C.F. of the given numbers.

The H.C.F. of $237$ and $81$ is $3$

Step 5: From Step 3, we have $3=75–6\times 12$ ......(i)

From Step 2: $6=81–75\times 1$.....(ii)

Substitute equation(ii) in equation(i), we, get $3=75–(81–75\times 1)\times 12$

$\Rightarrow 3=75–(81\times 12–75\times 12)$
$=75-81\times 12+75\times 12$

Therefore, $3=75\times 13–81\times 12$ .....(iii)

From Step 1, $75=237–81\times 2$.......(iv)

Substitute equation(iv) in equation (iii), we get

$3=(237–81\times 2)\times 13-(81\times 12)$
$\Rightarrow 3=(237\times 13–81\times 26)–(81\times 12)$
$\Rightarrow 3=(237\times 13)–(81\times 38)$
Thus, H.C.F. of $237$ and $81=(237\times 13)+(81\times (–38\left)\right)$

Therefore, $(237\times 13)+(81\times (–38\left)\right)$ is the representation of H.C.F. of $237$ and $81$ as linear combination of $237$ and $81$.