Finding All Factors of 18

Find all the factors of 18.

Start with 1: $18 \div 1 = 18$, so 1 and 18 are factors = Factors: $1, 18$

Try 2: $18 \div 2 = 9$, so 2 and 9 are factors = Factors: $1, 2, 9, 18$

Try 3: $18 \div 3 = 6$, so 3 and 6 are factors = Factors: $1, 2, 3, 6, 9, 18$

Try 4 and 5: $18 \div 4 = 4.5$ (not whole), $18 \div 5 = 3.6$ (not whole) = 4 and 5 are NOT factors

Stop at the square root: $\sqrt{18} \approx 4.2$, so we've found all factors = All factors: $1, 2, 3, 6, 9, 18$

Answer: The factors of 18 are: $1, 2, 3, 6, 9, 18$

Listing Multiples

List the first five multiples of 7.

Multiply 7 by 1: $7 \times 1 = 7$ = First multiple: $7$

Multiply 7 by 2: $7 \times 2 = 14$ = Second multiple: $14$

Multiply 7 by 3: $7 \times 3 = 21$ = Third multiple: $21$

Multiply 7 by 4: $7 \times 4 = 28$ = Fourth multiple: $28$

Multiply 7 by 5: $7 \times 5 = 35$ = Fifth multiple: $35$

Answer: The first five multiples of 7 are: $7, 14, 21, 28, 35$

Factor Pairs

Find all factor pairs of 36.

Start with 1: $1 \times 36 = 36$ = Factor pair: $(1, 36)$

Try 2: $2 \times 18 = 36$ = Factor pair: $(2, 18)$

Try 3: $3 \times 12 = 36$ = Factor pair: $(3, 12)$

Try 4: $4 \times 9 = 36$ = Factor pair: $(4, 9)$

Try 6: $6 \times 6 = 36$ = Factor pair: $(6, 6)$

Answer: The factor pairs of 36 are: $(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)$

Is It a Multiple?

Is 72 a multiple of 8?

Understand the question: We need to check if $72 = 8 \times$ some whole number = Check: is 72 divisible by 8?

Divide to check: $72 \div 8 = 9$ = Result is a whole number

Verify: $8 \times 9 = 72$ ✓ = Yes, 72 is a multiple of 8

Answer: Yes, 72 is a multiple of 8 because $8 \times 9 = 72$

Mistake: Confusing factors and multiples

Why: The words sound similar but mean opposite things. Factors divide INTO a number; multiples are made BY multiplying.

Correct: Remember: Factors are Few (limited), Multiples are Many (infinite). 12 has 6 factors, but infinite multiples.

Mistake: Forgetting that 1 is always a factor

Why: Every number divided by 1 gives itself, so 1 is always a factor.

Correct: Always start your factor list with 1, and don't forget the number itself is also a factor.

Mistake: Thinking 0 is a multiple of every number

Why: While $n \times 0 = 0$ is true, we usually start listing multiples from the number itself.

Correct: The first multiple of any number $n$ is $n$ itself (not 0). So multiples of 5 start: $5, 10, 15, \ldots$

Mistake: Missing factor pairs

Why: When finding factors, students sometimes forget that factors come in pairs.

Correct: When you find one factor, find its pair: if $18 \div 2 = 9$, then both 2 AND 9 are factors.

Party Planning

When organizing events, you need to divide things equally among guests.

You have 30 cupcakes. Which group sizes can share them equally? The factors of 30: $1, 2, 3, 5, 6, 10, 15, 30$. So you can have groups of 2, 3, 5, 6, 10, 15, or 30 people.

Scheduling and Patterns

Multiples help us find when events occur together.

Bus A comes every 15 minutes. Bus B comes every 20 minutes. When do they arrive together? Find common multiples of 15 and 20: $60, 120, 180, \ldots$ They arrive together every 60 minutes.

Arranging Objects

Factors help determine how to arrange items in rows and columns.

You have 48 chairs to arrange. Possible arrangements: 1 row of 48, 2 rows of 24, 3 rows of 16, 4 rows of 12, 6 rows of 8, 8 rows of 6, etc.

Q: Is a number a factor and multiple of itself?

A: Yes! Every number is both a factor of itself ($12 \div 12 = 1$) and a multiple of itself ($12 \times 1 = 12$).

Q: How do I know when I've found all the factors?

A: Stop checking when your divisor passes the square root. For 36, $\sqrt{36} = 6$, so after checking up to 6, you've found all factor pairs.

Q: What's the difference between factors and divisors?

A: They mean the same thing! 'Factor' and 'divisor' are two words for numbers that divide evenly into another number.