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Let $ϕ$ be the Euler phi-function . Let $p$ and $q$ two different prime numbers.

Prove that $p$ divides $ϕ(q^p) − ϕ(q)$. ${}{}{}{}{}$

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up vote 4 down vote accepted


$$\text{As }q\text{ is prime }\phi(q^p)=q^{p-1}(q-1)$$

$$\text{So, }\phi(q^p)-\phi(q)=q^{p-1}(q-1)-(q-1)=(q-1)(q^{p-1}-1)$$

Use Fermat's Little Theorem

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