Question

If $d$ is the HCF of $30$ and $72$ with a linear equation $d=30x+72y$ find $x$ and $y$.

Answer 1

Applying Euclid's division lemma to $30$ and $72$

Since $72>30$

$72=30\times 2+12$ ....... $\left(1\right)$

$30=12\times 2+6$ ...... $\left(2\right)$

$12=6\times 2+0$ ...... $\left(3\right)$

The remainder has now become zero,

Since the divisor at this stage is $6$

The HCF of $30$ and $72$ is $6$

Now from $\left(2\right)$, we have

$30=12\times 2+6$

Rearrange this

$6=30-12\times 2$

$\Rightarrow 6=30-\left[\right(72-30\times 2)\times 2]$ { from ( 1 ) }

$\Rightarrow 6=30-72\times 2+4\times 30$ [ using distributive property ]

$\Rightarrow 6=30\times \left(5\right)+72\times (-2)$ ...... $\left(4\right)$

According to the problem,

$d=30x+72y$ ........ $\left(5\right)$

Comparing $\left(4\right)$ and $\left(5\right),$ we get

$x=5$ and $y=-2$