Question

If a,b,c are integers and p is prime that divides both a and $a+bc$, prove that $p|b$or $p|c$.

Short answer

It is proved that if p is a prime that divides both a and$a+bc$ , then$p|b$ and $p|c$ .

Step-by-step solution

Consider

It is given that p is a prime that divides both a and $a+bc$, this means that $p|\left(a+bc\right)$and $p|a$. By division algorithm we havethe following:

$a=pk$and$a+bc=pl$

This implies,

role="math" localid="1646204473090" $\begin{array}{rcl}pk+bc& =& pl\\ bc& =& pl-pk\\ bc& =& p\left(l-k\right)\end{array}$

Therefore,$p|bc$.

p|cStep 2: Show the required result 

Then according tothetheorem, an integer p is prime if and only if p has the property whenever $p|bc$, then $p|b$or .

This implies that$p|b$ and $p|c$.

Hence,it is provedthat if p is a prime that divides both a and$a+bc$ , then $p|b$and $p|c$.