Indeed, the definition of a prime number is to be divisible by two distinct integers, 1 and itself.
Concerning the number 1, the two divisors 1 and itself are not distinct: they are equal.
Therefore, 1 does not match the definition of a prime number, and therefore is not a prime number!
Parity of 1
One is of course an odd number. It is the smallest non-negative odd number.
What is the square root of 1?
One is its own square root, and its own square number. That is, 1 squared equals 1, and the square root of 1 equals 1 too.
What are the multiples of 1?
All integers are multiples of 1. Indeed, a number is a multiple of 1 if it is divisible by 1. There are therefore infinitely many multiples of 1 (including 0 and 1, of course). The smallest multiples of 1 are:
1 is a single-digit number, because it is strictly less than 10; 1 is in fact itself a digit.
How to determine whether an integer is a prime number?
To determine the primality of a number, i.e., know whether it is a prime number, several algorithms can be used.
The most naive technique is to test all divisors strictly smaller to the number of which we want to determine the primality.
First, we can eliminate all even numbers greater than 2.
Then, we can stop this check when we reach the square root of the number of which we want to determine the primality.
Historically, the sieve of Eratosthenes (dating from the Greek mathematics) implements this technique in a relatively efficient manner.
More modern techniques include the sieve of Atkin, probabilistic algorithms, and the cyclotomic AKS test.