Question

If p and q are distinct primes such that p|c, and q|c, prove that pq|c. [Hint if$\mathbf{c}=\mathbf{pk}$ , then $\mathbf{q}\mathbf{|}\mathbf{pk}$, use Theorem 1.5 ]

Short answer

The proof is obtained by writing c as the multiple of p and multiple of q such that p, q are relatively prime.

Step-by-step solution

Determine relatively prime relation

Consider p|c implies $\mathbf{c}\mathbf{=}\mathit{p}\mathbf{k}$then, q|c implies q|pk as both the variables are prime then they are distinct and are also relatively prime. That is q|k

Prove the given relation:

Since, $k=qm$,

Substitute this in the relation $c=pk$ as:

$\begin{array}{l}c=pqm\\ \therefore pq|c\end{array}$

Therefore, the proof is obtained by writing c as the multiple of p and multiple of q such that p, q are relatively prime.