$np$ is the order of $n$ in $\mathbb Z_n$. Why is it true?

I understand that $\phi(n)$ divides $n$, and that $\phi(p)=p-1$. But why does $p \mid n$, and $n = 1 \cdot p + 1 \cdot p^2 + 1 \cdot p^3 +…$ imply that $n \mid np$?

A:

Let $n=p^am$ where $a\ge 1$. Then $n\mid np$.

A:

$$n=\underbrace{1\cdot p+1\cdot p^2+1\cdot p^3+\cdots+1\cdot p^m}_m$$
$$=\underbrace{1\cdot p}_{=p-1}\cdot m+\underbrace{1\cdot p^2}_{=p}\cdot m+\underbrace{1\cdot p^3}_{=p^2-1}\cdot m+\cdots+\underbrace{1\cdot p^m}_{=p^{m-1}-1}\cdot m$$
$$\implies np=\underbrace{1\cdot p}_{=p-1}\cdot m^2+\underbrace{1\cdot p^2}_{=p}\cdot m^2+\underbrace{1\cdot p^3}_{=p^2-1}\cdot m^2+\cdots+\underbrace{1\cdot p^{m-1}}_{=p^{m-2}-1}\cdot m^2$$

A:

When we say that $\phi(n)$ divides $n$ we mean that $\varphi(n) = \prod_{i=1}^{\infty} p_i^{k_i}$ where $p_i$ is the $i$th prime and $\prod_{i=1}^{\infty} p_i^{k_i} = \varphi(n)$.
We know that \$

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The cardiac field is currently using catheter-based medical procedures for a variety of treatments. These treatments include the use of balloon or basket catheters, stent delivery catheters, electrophysiology (EP) catheters, etc. In a common method of treatment, a catheter is inserted into a patient””s vasculature and guided to the site where a medical procedure is to be performed. The catheter may be guided by a hollow needle, known as a trocar, or a catheter may simply be inserted into a blood vessel. The trocar is then removed, leaving the catheter in place. The catheter may then be equipped with one or more medical devices, which are used to complete the medical procedure. For example, the catheter may be used to deliver treatment agents, such as therapeutic agents, drugs, etc., to a preselected location, in the subject. The catheter may also be used to deliver diagnostic agents to a preselected