Question

Show that, for any integers $p$and $q$( $q$not equal to zero).

$\left(p-1\right)-\frac{p(q-1)}{q}=\frac{p}{q}-1$

What does this equation have to do with the current section?

Short answer

The given proof has shown.

Step-by-step solution

Step 1. Given information:

$p$and $q$are integers

$q$is not zero.

We have to prove:

$\left(p-1\right)-\frac{p(q-1)}{q}=\frac{p}{q}-1$

Step 2. Proof of statement.

$$\begin{gathered} L.H.S=\left(p-1\right)-\frac{p(q-1)}{q} \\ =(p-1)-\left(\frac{pq-p}{q}\right) \\ =(p-1)-\left(\frac{pq}{q}-\frac{p}{q}\right) \\ =\left(p-1\right)-\left(p-\frac{p}{q}\right) \\ =p-1-p+\frac{p}{q} \\ =\frac{p}{q}-1 \end{gathered}$$

$R.H.S.=\frac{p}{q}-1$

Here $L.H.S.=R.H.S.$

So given statement is proved.