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Mathematics › 🎯 Grade 4 › Factors, Multiples & Prime Numbers: Arranging the Tiles
🎯 Grade 4 · Lesson 8 of 9

Factors, Multiples & Prime Numbers: Arranging the Tiles

A factor is any number that tiles the rectangle perfectly. A prime number only tiles in one single row.

Grade 4Elementary
Factors, Multiples & Prime Numbers: Arranging the Tiles — illustration
💡
The big idea: Factors and multiples are two sides of the same coin. If 3 × 4 = 12, then 3 and 4 are both factors of 12, and 12 is a multiple of both 3 and 4. Arranging tiles into rectangles makes this concrete: every valid rectangular arrangement reveals a factor pair. A prime number can only form a single rectangle (1 × itself), which is why it has exactly two factors.
🎯 By the end, you'll be able to
  • List all factor pairs of a number up to 100
  • Identify whether a number is prime or composite
  • Generate multiples of a given number
  • Distinguish between factors (what divides in) and multiples (what a number divides into)
📎 You should already know
  • Multiplication facts up to 10×10
  • Division with remainders

Factor pairs from rectangles

Take 12 square tiles and try every possible rectangle you can make without gaps or overlaps. You'll find: 1×12, 2×6, 3×4 (and their reverses 12×1, 6×2, 4×3). Each dimension used is a factor of 12. So the factors of 12 are: 1, 2, 3, 4, 6, 12.

To find factor pairs systematically, start at 1 and work up to the square root. Once your trial divisor exceeds √n, you've already found every unique pair (because beyond that point you'd just be repeating them in reverse).

🎮 Factor Array Explorer LIVE
Use the slider to set the total number of tiles. Try different row counts — the widget snaps only to whole-number arrangements, revealing which row counts are valid factors. All valid factor pairs appear in the readout.
🔑 Prime vs composite
  • Prime: exactly 2 factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13. Only one rectangle is possible (a single row).
  • Composite: more than 2 factors. Examples: 4, 6, 8, 9, 12. Multiple rectangles exist.
  • Special case: 1 has only 1 factor (itself) — it is neither prime nor composite.
✨ Factors vs multiples — the direction matters

Factors are smaller (or equal) — they divide INTO the number. 3 is a factor of 12 because 12 ÷ 3 = 4 (no remainder).

Multiples are larger (or equal) — the number divides INTO them. 12 is a multiple of 3 because 3 × 4 = 12. The multiples of 3 are: 3, 6, 9, 12, 15, 18, …

Memory hook: Factors are Few (at most n of them), Multiples are Many (infinitely many).

📝 Worked example: List all factors of 24 and state whether 24 is prime or composite.
  1. Test divisors from 1 up to √24 ≈ 4.9 (so up to 4).
  2. 1: 24 ÷ 1 = 24 ✓ → pair (1, 24)
  3. 2: 24 ÷ 2 = 12 ✓ → pair (2, 12)
  4. 3: 24 ÷ 3 = 8 ✓ → pair (3, 8)
  5. 4: 24 ÷ 4 = 6 ✓ → pair (4, 6)
  6. 5: 24 ÷ 5 = 4.8 ✗ (not whole)
  7. Factors: 1, 2, 3, 4, 6, 8, 12, 24 — 8 factors.
✓ 24 is <strong>composite</strong> (more than 2 factors).
📝 Worked example: Is 37 prime or composite?
  1. √37 ≈ 6.1, so test divisors 2, 3, 4, 5, 6.
  2. 37 ÷ 2 = 18.5 ✗
  3. 37 ÷ 3 = 12.33... ✗
  4. 37 ÷ 4 = 9.25 ✗
  5. 37 ÷ 5 = 7.4 ✗
  6. 37 ÷ 6 = 6.16... ✗
  7. No divisor works up to √37 — only factors are 1 and 37.
✓ 37 is <strong>prime</strong>.

Check your understanding

1. Which of the following is a factor of 36?
36 ÷ 9 = 4 exactly. So 9 is a factor. 36 ÷ 8 = 4.5, 36 ÷ 14 and ÷ 16 are not whole numbers.
2. How many factors does 16 have?
Factors of 16: 1, 2, 4, 8, 16 — that's 5 factors. (Note: 4 × 4 = 16, so 4 is paired with itself — count it once.)
3. Which number is prime?
9 = 3×3, 15 = 3×5, 21 = 3×7. Only 23 has no factors other than 1 and 23 — it's prime.
4. List the first five multiples of 7.
Multiples of 7: 7×1=7, 7×2=14, 7×3=21, 7×4=28, 7×5=35.
5. Sam says 'all even numbers greater than 2 are composite.' Is he right?
Any even number n > 2 is divisible by 1, 2, and n itself — at least 3 factors, so it's composite. 2 is the only even prime.
✅ Key takeaways
  • A factor divides into a number with no remainder; factors come in pairs (a × b = n).
  • Find all factor pairs by testing divisors from 1 up to the square root of the number.
  • A prime number has exactly 2 factors (1 and itself); a composite number has more than 2.
  • The number 1 is neither prime nor composite — it's a special case.
  • Multiples are what you get by multiplying: multiples of 4 are 4, 8, 12, 16, … — infinitely many.