Factor Pairs, Multiples, and Prime or Composite Numbers

Finding a Number's Complete Structure

In this lesson:

• Find ALL factor pairs of a number using a systematic method
• Test whether a number is a multiple of a one-digit number, quickly
• Classify whole numbers as prime, composite, or neither

Learning Objectives for Today's Lesson

By the end of this lesson, you should be able to:

1. Find all factor pairs of a number systematically
2. Explain the factor-multiple duality
3. Use divisibility tests for 2, 3, 5, and 9
4. Classify numbers as prime, composite, or neither
5. Use rectangles to model factor pairs

How Many Rectangles Can You Build?

"I have 24 square tiles. I want a perfect rectangle — all rows the same length, no gaps. How many different rectangles can I make?"

Try it — how many can you find?

How Can You Be Sure You're Done?

Most people find 1×24, 2×12, and 3×8 — but miss 4×6.

Guessing finds some rectangles. It doesn't prove you found all of them.

Today's tool: a systematic method that checks every possibility in order.

The Systematic Method: Testing 24

Test 1, 2, 3, 4: all succeed — pairs 1×24, 2×12, 3×8, 4×6

Test 5: 24 ÷ 5 leaves a remainder — not a factor

Test 6: 24 ÷ 6 = 4 — already found! Stop here.

Is 6 a Multiple of 42?

Since 6 × 7 = 42:

Is 6 a factor of 42, or a multiple of 42? Predict first.

A factor goes INTO a number. A multiple is BUILT FROM one.

6 is a factor of 42. 42 is a multiple of 6.

Second Example: All Factor Pairs of 36

36 ÷ 1 = 36, 36 ÷ 2 = 18, 36 ÷ 3 = 12, 36 ÷ 4 = 9

36 ÷ 6 = 6 — the two factors are equal! 36 is a perfect square.

Five factor pairs: 1×36, 2×18, 3×12, 4×9, 6×6

Your Turn: All Factor Pairs of 30

Test every number starting at 1. Write down every result, even the failures.

Where do the factors start to repeat? That's your stopping point.

Practice: Factor Pairs and the Duality

Find all factor pairs for each. For two of them, also answer: is a a factor of b? Is b a multiple of a?

18, 40, 48, 64, 72, 100

Faster Ways to Check Divisibility?

You divide by 2, then 3, then 4, every single time.

Are there shortcuts that tell you the answer before you divide?

Yes — divisibility rules let you check some numbers just by looking at the digits.

Divisibility Rules for 2, 5, 9, and 10

• By 2: last digit even
• By 5: last digit 0 or 5
• By 10: last digit 0
• By 3: digit sum divisible by 3 — 84's sum, 12, works
• By 9: digit sum divisible by 9 — 81's sum, 9, works; 84's doesn't

Applying the Rules to 72

72 ÷ 8 = 9 — no remainder. Yes, a multiple of 8.

72 ÷ 7 = 10 R 2 — remainder. No, not a multiple of 7.

For 4, 7, and 8, there's no digit trick — just divide directly.

Circling Back: Factor Pairs of 60, Faster

Tests 2 through 6 all succeed using the shortcuts (even, digit sum, clean division, ends in 0)

Test 7 fails; Test 8 fails too (60 ÷ 8 = 7 R 4) — quotient smaller than divisor, so stop

Six factor pairs: 1×60, 2×30, 3×20, 4×15, 5×12, 6×10

Practice: Multiples and Factor Pairs

Is n a multiple of d?

1. Is 84 a multiple of 6? 2. Is 91 a multiple of 3? 3. Is 45 a multiple of 9?

Find all factor pairs, using the shortcuts:
4. 50 (ends in 0) 5. 45 (ends in 5)

Numbers With Very Few Factor Pairs

Find all factor pairs of 12, 7, and 1.

• 12: 1×12, 2×6, 3×4 — three pairs
• 7: 1×7 — one pair only
• 1: 1×1 — just one factor

Some numbers have many factor pairs. Some have almost none. Why?

Defining Prime, Composite, and Neither

• Prime: exactly two distinct factors, 1 and itself — like 7
• Composite: more than two factors — like 12
• 1 is neither: it has only ONE factor, so it can't have two distinct ones

2 Is the Only Even Prime

Factor pairs of 2: just 1 × 2. Exactly two factors — prime.

Every other even number is divisible by 2, giving it at least three factors: 1, 2, and itself. Composite.

Being even usually means composite — 2 is the single exception.

One Test You Can Always Run

Run the systematic search. If 1 × n is the only pair that works, the number is prime.

Preview: to find every prime under 100, cross out multiples of 2 (except 2), then 3, then 5, then 7 — whatever survives is prime.

Practice: Classify Ten Different Numbers

Prime, composite, or neither? Justify with a factor pair (except for primes and 1).

11, 15, 23, 27, 2, 49, 51, 1, 37, 44

A bare label isn't an answer — show the factor pair that proves it.

Prove It: Composite, Prime, and a Range

1. Prove 51 is composite: name a factor other than 1 and 51.
2. Explain why 37 is prime: what did you test, and what happened?
3. List all primes between 40 and 60.

Four Errors Worth Watching For

1. 1 is not prime: only ONE factor — prime needs two
2. Skipped factors: test every number in order; record each result
3. Factor vs. multiple: a factor goes INTO a number, a multiple comes FROM one
4. Odd ≠ prime: still test 3, 5, 7

Key Takeaways: One Search, Three Views

Finding factor pairs, divisibility, and prime/composite are one search, viewed three ways.

✓ Test upward from 1; stop when the factors switch places

✓ A factor goes into a number; a multiple is built from one

✓ Count the pairs: one means prime, more means composite

What's Next: Factors in Fractions and Beyond

• 4.NF.A.1 (next in this course): common factors help simplify and compare fractions
• Grade 6: greatest common factor, least common multiple, and prime factorization all build directly on today's work

Everything you learned about factor pairs and primes becomes a tool, not just a topic.