# Science:Math Exam Resources/Courses/MATH312/December 2008/Question 06 (a)

MATH312 December 2008

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### Question 06 (a)

The purpose of this problem is to prove the following theorem.

Theorem 1. For all positive integers ${\displaystyle \displaystyle a,m}$ we have ${\displaystyle \displaystyle a^{m}\equiv a^{m-\phi (m)}\mod {m}}$

Let ${\displaystyle \displaystyle a,m}$ be positive integers. For ${\displaystyle \displaystyle m=1}$, the theorem holds trivially, so we assume from now on that ${\displaystyle \displaystyle m>1}$ and write its prime-power factorization as ${\displaystyle \displaystyle m=p_{1}^{e_{1}}\cdot ...\cdot p_{k}^{e_{k}}}$ for different primes ${\displaystyle \displaystyle p_{1},...,p_{k}}$ and positive integer exponents ${\displaystyle \displaystyle e_{1},...,e_{k}}$ and some positive integer k. Let ${\displaystyle \displaystyle i\in \{1,..,k\}}$ and focus on the prime power ${\displaystyle \displaystyle p_{i}^{e_{i}}}$ in the prime-power factorization of m.

(a) Prove that if ${\displaystyle \displaystyle p_{i}\nmid a}$ then ${\displaystyle \displaystyle p_{i}^{e_{i}}\mid (a^{\phi (m)}-1)}$

Hint: Use Euler's theorem and prove/use that ${\displaystyle \displaystyle \phi (p_{i}^{e_{i}})\mid \phi (m)}$

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