Multiples

Numerator (top number) and Denominator (bottom number) are used to form a fraction or division problem. The video stresses the ideas of factors and multiples using these terms.
Factor: If the result of the division is an integer, then the bottom number is a factor of the top number. In symbols: if $\dfrac{N}{D} \in \mathbb{Z}$ , then $D \mid N$ (D is a factor of N).
Multiple: The reverse relationship of factors. If the division yields an integer, the top number is a multiple of the bottom number, because the bottom number times some integer equals the top number.

Example illustrating a factor: 30 over 10 equals 3, both integers. This means that 10 is a factor of 30. In symbols: $\dfrac{30}{10} = 3 \quad \text{and} \quad 30 = 10 \cdot 3.$ Thus, 10 is a factor of 30.
Alternative phrasing: We can say that 30 is divisible by 10, or that 30 is a multiple of 10.
Formal view: If a fraction (\dfrac{N}{D}) is an integer, then there exists a positive integer (k) such that $N = D \cdot k.$ This is the core idea of divisibility and factors.

Multiples are the results of multiplying a number by positive integers. The fundamental process is the same as the reverse of factoring.
Example with 30 and 10: 30 is a multiple of 10 because $30 = 10 \cdot 3.$ (Equivalently, $\dfrac{30}{10} = 3 \in \mathbb{Z}$ .)
Example with 5:
- The multiples of 5 are obtained by multiplying 5 by positive integers: $5 \times 1 = 5, \quad 5 \times 2 = 10, \quad 5 \times 3 = 15, \quad 5 \times 4 = 20.$ The green values in the video are 5, 10, 15, 20, which are multiples of 5.
- The multiplication process can continue indefinitely: 5 \times n, \quad n \in \mathbb{Z}_{>0}, and you can extend to five times 5, 6, 7, …, all the way to infinity.

Zero as a multiple:
- For any integer (a), (a * 0 = 0). Therefore, 0 is a multiple of every integer. In symbols: $a \cdot 0 = 0 \Rightarrow 0 \text{ is a multiple of } a.$
- Complementary idea: 1 is a factor of every integer. In symbols: $1 \mid a.$ (This means a is divisible by 1.)

Negative multiples exist because multiplication by negative integers yields negative results. Example: $5 \times (-2) = -10.$ Hence, (-10) is a multiple of (5).
The video notes that while negative multiples exist, the GRE and similar tests typically emphasize positive multiples or positive factors.
The video also mentions that problems about zero as a multiple or about negative factors are usually delineated or not emphasized in standard tests.

If the division of the top by the bottom yields an integer, the bottom is a factor of the top. Example: $\dfrac{30}{10} = 3 \Rightarrow 10 \text{ is a factor of } 30.$ And equivalently, $30 = 10 \cdot 3.$ Thus, 30 is a multiple of 10.
Conversely, if a number is a multiple of another, it can be expressed as the product of the divisor and a positive integer. Example: $30 = 10 \cdot 3.$
Multiples can be generated by multiplying the chosen base number by positive integers: for base 5, the multiples are $5, 10, 15, 20, \ldots$ (i.e., all numbers of the form 5 \cdot n, \ n \in \mathbb{Z}_{>0}).
Special cases:
- Zero is a multiple of every integer (since any number times 0 gives 0).
- One is a factor of every integer (since every integer divided by 1 equals itself).
- Negative multiples exist but are less commonly tested in standard problem sets.

These concepts underpin divisibility tests, simplification of fractions, and factorization problems.
Understanding that factors are the numbers you can multiply by to obtain the top number, while multiples are the results of multiplying the base by positive integers, helps in recognizing when fractions simplify, when numbers divide each other, and how to generate sequences of multiples for quick mental math.
Foundational perspective: factors are connected to divisibility; multiples are the forward-building process from a base number. The two views are inverses of each other.

Divisibility relation (factor): D \mid N \quad\text{iff}\quad \exists k \in \mathbb{Z}_{>0} \text{ such that } N = D \cdot k.
Integer quotient condition: $\dfrac{N}{D} \in \mathbb{Z}.$
Multiple definition (based on base number B): { B \cdot n \mid n \in \mathbb{Z}_{>0} }.
Example identities:
- $30 = 10 \cdot 3.$
- $\dfrac{30}{10} = 3.$
- For base 5: {5, 10, 15, 20, \ldots} = {5 \cdot n \mid n \in \mathbb{Z}_{>0}}.