Number Sense and Operations - Factors and Multiples

Factors and Factor Pairs: Factors are whole numbers that can be multiplied together to result in another number. For example, the factors of $12$ are $1, 2, 3, 4, 6,$ and $12$. To visualize this, imagine a 'Factor Rainbow' where arcs connect pairs of numbers that multiply to the target: $1$ is linked to $12$, $2$ to $6$, and $3$ to $4$.

Multiples: A multiple of a number is the product of that number and any whole number. The multiples of $4$ are $4, 8, 12, 16, \dots$. Visually, you can represent multiples as jumps on a number line, where each jump is of equal length, landing on the next multiple in the sequence.

Prime and Composite Numbers: A prime number has exactly two factors: $1$ and itself (e.g., $2, 3, 5, 7$). A composite number has more than two factors (e.g., $4, 6, 8, 9, 10$). Visually, prime numbers cannot be formed into rectangular arrays other than a single line of dots, whereas composite numbers can be arranged into multiple rectangular shapes (e.g., $6$ dots can be $1 \times 6$ or $2 \times 3$).

Divisibility Rules: These are mental math shortcuts to determine if a number is divisible by another without full division. For example, a number is divisible by $5$ if it ends in $0$ or $5$, and divisible by $3$ if the sum of its digits is a multiple of $3$. Imagine a checklist or flowchart used to quickly 'filter' numbers based on their ending digits or digital sums.

Greatest Common Factor (GCF): The GCF is the largest factor shared by two or more numbers. In a Venn Diagram, if you place the factors of one number in the left circle and the factors of another in the right, the GCF is the largest number found in the middle 'intersection' where the circles overlap.

Least Common Multiple (LCM): The LCM is the smallest common multiple shared by two or more numbers. If you visualize two people running around a track at different speeds, the LCM represents the exact time or distance at which both runners would cross the starting line at the same moment.

Prime Factorization and Factor Trees: Every composite number can be expressed as a product of prime numbers. A 'Factor Tree' is a visual tool where the main number is the top 'trunk', and it branches out into pairs of factors. The process stops when every branch ends in a 'leaf' that is a prime number (e.g., $60 = 2 \times 2 \times 3 \times 5$).