Question

If a,b,c,d are integers and p is a primeof $a-b$and $c-d$, prove that is a prime factor of$\left(a+c\right)-\left(b+d\right)$ .

Short answer

It is proved that $a-b$is a factor of $\left(a+c\right)-\left(b+d\right)$.

Step-by-step solution

Use the given part

It is given that, p is a prime factor of $a-b$and $c-d$.

Then $a-b$and $c-d$can be written as follows:

$$\begin{gathered} a-b=pk \\ c-d=ps \end{gathered}$$

Prove p is a factor of a+c-b+d

The expression$\left(a+c\right)-\left(b+d\right)$ can be written as $\left(a-b\right)+\left(c-d\right)$.

Now, use$a-b=pk$ and$c-d=ps$ in $\left(a-b\right)+\left(c-d\right)$.

$\begin{array}{rcl}\left(a-b\right)+\left(c-d\right)& =& pk-ps\\ & =& p\left(k-s\right)\end{array}$

This implies p is a factor of$\left(a-b\right)+\left(c-d\right)$or$\left(a+b\right)-\left(c+d\right)$ .