1. Definition of prime numbers and some basic properties An integer $n$ greater than 1 has at least two distinct positive divisors: 1 and $n$. If $n$ has no divisor greater than 1 and less than $n$, then $n$ is called a prime number. If $n$ has a divisor greater than 1 and less than $n$, i.e., $n$ can be expressed in the form of $a \cdot b$, then $n$ is called a composite number.
(1) Any integer greater than 1 must have a prime factor.
(2) There is only one positive integer that is both even and prime, which is 2.
(3) Let $p$ be a prime number, and $n$ be any integer, then either $p$ divides $n$, or $p$ and $n$ are coprime.
(4) Let $p$ be a prime number, and $a, b$ be integers. If $p \mid a b$, then at least one of $a, b$ is divisible by $p$. Proof: If $p$ does not divide $a$ and $b$, then $p$ is coprime with $a$ and $b$, thus $p$ is coprime with $a b$, which is a contradiction!
(5) There are infinitely many prime numbers. Prove this proposition by contradiction, assume there are only finitely many prime numbers, let $p_{1}=2, p_{2}$ $=3, \cdots, p_{k}$ be all the prime numbers, consider $N=p_{1} p_{2} \cdots p_{k}+1$, clearly $N>1$. Therefore, $N$ has a prime factor $p$. Since $p_{1}, p_{2}$, $\cdots, p_{k}$ are all the prime numbers, $p$ must equal some $p_{i}(1 \leqslant i \leqslant k)$, thus $p$ divides $N-p_{1} p_{2} \cdots p_{k}=1$, which is impossible, therefore there are infinitely many prime numbers.