Playing with Numbers - Prime and Composite Numbers

Factors and Multiples: A factor of a number is an exact divisor of that number. For example, the factors of $6$ are $1, 2, 3,$ and $6$. Visually, if you have $6$ beads, you can arrange them in perfect rectangular groups of $1 \times 6$, $2 \times 3$, $3 \times 2$, or $6 \times 1$.

Prime Numbers: These are numbers that have exactly two factors: $1$ and the number itself. Examples include $2, 3, 5, 7, 11,$ and $13$. Visually, a prime number of objects (like $5$ stars) can only be arranged in a single row or a single column; they cannot form a multi-row rectangle.

Composite Numbers: Numbers having more than two factors are called composite numbers. For example, $4, 6, 8, 9, 10$ are composite. Visually, composite numbers can always be arranged in at least one rectangular array that has more than one row and one column, such as $9$ dots arranged in a $3 \times 3$ square.

The Number 1: The number $1$ is a unique case in mathematics. It has only one factor (itself). Therefore, it is classified as neither a prime number nor a composite number.

Sieve of Eratosthenes: This is a systematic visual method to find all prime numbers up to a given limit. Imagine a grid of numbers from $1$ to $100$. By crossing out $1$, circling $2$ and crossing out all its multiples, then circling $3$ and crossing out all its multiples, and repeating for the next available numbers, the circled numbers remaining on the grid are the primes.

Even and Odd Numbers: All multiples of $2$ are even numbers ($2, 4, 6, 8 \dots$), and the rest are odd numbers ($1, 3, 5, 7 \dots$). Notably, $2$ is the smallest prime number and the only even prime number; every other prime number is odd.

Twin Primes: Two prime numbers are called twin primes if there is only one composite number between them, or their difference is $2$. Examples include $(3, 5)$, $(5, 7)$, and $(11, 13)$. On a number line, these appear as pairs of primes separated by exactly one even number.