Properties of Multiples

1. Definition: A multiple of a number \(n\) is any number that can be written as \(n \times k\), where \(k\) is an integer.

2. Multiplication by Zero: Every number has zero as a multiple because \(n \times 0 = 0\).

3. Infinite Set: Every number has infinitely many multiples, as you can always multiply by larger integers to create new multiples (e.g., \(n, 2n, 3n, \dots\)).

4. Common Multiples: Two or more numbers may share common multiples. The least common multiple (LCM) is the smallest number that is a multiple of all given numbers.

5. Divisibility: A number \(m\) is divisible by \(n\) if and only if \(m\) is a multiple of \(n\). This is key for checking divisibility.

6. Multiple of 1: Every number is a multiple of 1, as \(n \times 1 = n\).

7. Even and Odd Multiples:

8. Even numbers have even multiples (e.g., \(2, 4, 6, \dots\)).

9. Odd numbers have both even and odd multiples depending on the value of \(k\).

10. Prime Numbers and Multiples: A prime number's only multiples, apart from itself, are generated by multiplying it by integers (e.g., multiples of 5: \(5, 10, 15, \dots\)).