Multiples and factors

Pick any whole number, say $6$. You can do two things with it. Multiply $6$ by $1, 2, 3, \dots$ to get its multiples ($6, 12, 18, 24, \dots$). Or divide other numbers by $6$ to ask which whole numbers go in evenly , to find what $6$ is a factor of. Multiples roll outwards. Factors look inwards.

Concept

A multiple of a number $n$ is anything of the form $n \times k$, where $k$ is a whole number $\geq 1$. The multiples of $5$ are:

$5, 10, 15, 20, 25, 30, 35, \dots$

There are infinitely many. The smallest positive multiple of $n$ is $n$ itself.

A factor of a number $n$ is a whole number that divides $n$ exactly, leaving zero remainder. The factors of $12$ are:

$1, 2, 3, 4, 6, 12$

Factors always come in pairs. For $12$: $1 \times 12$, $2 \times 6$, $3 \times 4$. List them by walking up from $1$ and pairing each with its partner: $1\leftrightarrow 12$, $2 \leftrightarrow 6$, $3 \leftrightarrow 4$. After $\sqrt{12} \approx 3.46$, you start repeating partners , so you can stop searching once you cross $\sqrt{n}$. Useful trick!

A handy connection: $a$ is a factor of $b$ if and only if $b$ is a multiple of $a$. So $4$ is a factor of $20$ exactly because $20$ is a multiple of $4$. Two ways of saying the same fact.

Special cases. Every number is a factor of itself ($n \times 1 = n$). Every number is a multiple of itself. And $1$ is a factor of every number ($1 \times n = n$). The number $1$ has only one factor: itself.

A nice picture: think of arranging $n$ pebbles into a rectangle. Every rectangle of $n$ pebbles corresponds to a factor pair. $12$ pebbles can be arranged as $1 \times 12$, $2 \times 6$, $3 \times 4$ , three rectangles, three factor pairs. A number is "very factorable" if it forms many rectangles. A number is "boring" if it forms only one rectangle ($1 \times n$) , those are the primes, which we will meet soon.

Worked examples

Example 1. List the first six multiples of $7$.

• $7 \times 1 = 7$, $7 \times 2 = 14$, $7 \times 3 = 21$, $7 \times 4 = 28$, $7 \times 5 = 35$, $7 \times 6 = 42$.
• Multiples: $7, 14, 21, 28, 35, 42$.

Example 2. Find all factors of $24$.

• Check $1$: yes. Partner $24$.
• Check $2$: $24 \div 2 = 12$ ✓. Partner $12$.
• Check $3$: $24 \div 3 = 8$ ✓. Partner $8$.
• Check $4$: $24 \div 4 = 6$ ✓. Partner $6$.
• Check $5$: $24 \div 5 = 4.8$ ✗.
• $\sqrt{24} \approx 4.9$. Stop.
• Factors: $1, 2, 3, 4, 6, 8, 12, 24$ (eight in all).

Example 3. Is $42$ a multiple of $7$? Is $7$ a factor of $42$?

• $42 \div 7 = 6$, no remainder. Yes to both.

Example 4. Find all factor pairs of $36$.

• $1 \times 36$, $2 \times 18$, $3 \times 12$, $4 \times 9$, $6 \times 6$.
• Factors: $1, 2, 3, 4, 6, 9, 12, 18, 36$.
• Notice $36$ has $9$ factors , an odd number, because $6$ is paired with itself. This always happens for squares.

Example 5. What is the smallest multiple of both $4$ and $6$?

• Multiples of $4$: $4, 8, 12, 16, 20, 24, \dots$
• Multiples of $6$: $6, 12, 18, 24, \dots$
• First common one: $12$. So the LCM of $4$ and $6$ is $12$.

Try it yourself

1. List the first $5$ multiples of $9$.
2. Find all factors of $30$.
3. Find all factors of $48$.
4. How many factors does $25$ have?
5. Is $84$ a multiple of $12$?
6. Find all factor pairs of $100$.
7. Find the smallest multiple of both $3$ and $5$.
8. Investigate: which numbers from $1$ to $20$ have an odd number of factors? What do you notice?

Activity

Rectangles game. With $24$ small beads, buttons, or pebbles, lay out every rectangle possible (so all of them are filled). Count each rectangle's two dimensions , those are the factor pairs of $24$. Now try with $36$, and then with $25$. Compare the number of rectangles each one makes. Which numbers make the fewest rectangles?