Proof Using Euler's Totient Function
The theorem can be also proved by using Euler's totient function φ. In this proof, the following property will be used:
Assume that there is only a finite number of primes. Call them p1, p2, ..., pr and consider the integer n = p1p2 ... pr. Any natural number a ≤ n has a prime divisor q. Then, q must be one of p1, p2, ..., pr, since that is the list of all prime numbers. Hence, for all a ≤ n, gcd(a, n) > 1 except if a = 1. This leads to φ(n) = 1, which contradicts the property above.
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