Factors and Multiples - Prime and Composite Numbers

Factors: Factors are the numbers that are multiplied together to get a product. For example, in the equation $2 \times 5 = 10$, both $2$ and $5$ are factors of $10$. You can visualize factors as the dimensions of a rectangular grid that can be formed using a specific number of unit squares.

Multiples: A multiple of a number is the product of that number and any non-zero whole number. For instance, the multiples of $4$ are $4, 8, 12, 16, \dots$. You can imagine this as 'skip counting' on a number line, where each jump is of the same length.

Prime Numbers: A prime number is a number greater than $1$ that has exactly two factors: $1$ and the number itself. Examples include $2, 3, 5, 7,$ and $11$. If you try to arrange $5$ blocks into a rectangle, you can only form a single row of $1 \times 5$ or a column of $5 \times 1$.

Composite Numbers: A composite number has more than two factors. For example, $4, 6, 8, 9,$ and $10$ are composite numbers. Unlike prime numbers, composite numbers like $12$ can be arranged into several different rectangular shapes, such as $2 \times 6$ or $3 \times 4$.

The Special Number 1: The number $1$ is unique because it has only one factor (itself). Because it does not have exactly two factors, it is not prime; because it doesn't have more than two factors, it is not composite. Therefore, $1$ is neither prime nor composite.

Even and Odd Prime Numbers: $2$ is the smallest prime number and it is the only even prime number. Every other even number (like $4, 6, 8, \dots$) is composite because it can be divided by $2$ in addition to $1$ and itself.

Factor Rainbows: A factor rainbow is a visual way to list all factors of a number in pairs. For the number $12$, you would draw an outermost arc connecting $1$ and $12$, an inner arc connecting $2$ and $6$, and the innermost arc connecting $3$ and $4$, showing that $1 \times 12 = 12$, $2 \times 6 = 12$, and $3 \times 4 = 12$.