Factors and Multiples - Divisibility Rules

Factors and Multiples: Factors are the numbers that divide a given number exactly without leaving a remainder. Multiples are the products obtained when a number is multiplied by other whole numbers. For example, if we have $5 \times 4 = 20$, then $5$ and $4$ are factors of $20$, and $20$ is a multiple of both $5$ and $4$.

Divisibility by 2, 5, and 10: These rules depend on the last digit of the number. A number is divisible by $2$ if the digit at the ones place is even ($0, 2, 4, 6, 8$). A number is divisible by $5$ if it ends in $0$ or $5$. A number is divisible by $10$ if the ones digit is exactly $0$. Imagine a number line where only numbers ending in these specific digits 'jump' into the divisibility group.

Divisibility by 3 and 9: These rules depend on the sum of the digits. For $3$, add all digits of the number; if the sum is a multiple of $3$, the number is divisible by $3$. Similarly, for $9$, if the sum of the digits is divisible by $9$, the whole number is divisible by $9$. Visualize 'breaking apart' the number into its individual digits and stacking them to see if the total height matches a multiple of $3$ or $9$.

Divisibility by 4 and 8: For $4$, check only the last two digits; if they form a number divisible by $4$, the whole number is. For $8$, check the last three digits. For example, in the number $1,524$, focus only on the '24' to determine divisibility by $4$.

Divisibility by 6: A number is divisible by $6$ if it satisfies the rules for both $2$ and $3$. This means the number must be even (ends in $0, 2, 4, 6, 8$) AND the sum of its digits must be a multiple of $3$.

Divisibility by 11: To check for $11$, find the sum of digits at odd places and the sum of digits at even places. Subtract the smaller sum from the larger sum. If the difference is $0$ or a multiple of $11$ (like $11, 22, 33$), the number is divisible by $11$. Visualize the digits in a 'see-saw' pattern, alternating between the left and right sides.

Relationship between Factors and Multiples: Every number is a factor of itself and $1$ is a factor of every number. A number is always greater than or equal to its factors, and a number is always less than or equal to its multiples (excluding zero).

Prime and Composite Numbers: A Prime number has exactly two factors ($1$ and itself), such as $2, 3, 5, 7$. A Composite number has more than two factors, such as $4, 6, 8, 9$. Note that $1$ is neither prime nor composite.