I need something explained or corrected:
In my number theory class we proved that there are an infinite number of primes using Euler's Phi Totient. It went something like this:
Let $M = p_1 p_2 \dots p_n$ be the product of all primes. Consider $1 < A \le M$:
Some prime must divide $A$, call it $q$. Since $q$ must be one of the primes, $q$ must divide $M.$
So the $gcd(A,M) > 1.$ Thus $\phi(M) = 1$ …?
Which is not even and contradicts the theorem that $\phi(N)$ is even for $N>2.$ Therefore there exists an infinite number of primes.
I get confused by the statement "Thus $\phi(M) = 1$"….
Did i possibly copy this proof down wrong? or am I missing something?
Thank you in advance.
Edit: by consider a such that…i meant consider an integer '$a$' I replaced it with $A$ to hopefully make it more clear.
Edit2: I'm sorry I am not familiar with using the equation editor. This is not a homework assignment, just studying for my exam. I just want to be able to understand this or clearify it.
Best Answer
By definition, $\phi(M)$ is the number of numbers in $S=\{1,2,\dots,M\}$ that are relatively prime with $M$. The argument shows that if $1<A\le M$, then $A$ is not relatively prime with $M$, so there is only one element of $S$ relatively prime with $M$, namely 1, and $\phi(M)$ is therefore equal to 1.