Factors , the building blocks
When 12 square patches can be arranged in a rectangle, the side lengths are the factors of 12. Factors are the numbers that divide a number exactly with no remainder.
What is this?
A factor of a number is a number that divides it exactly. Let us find the factors of 12:
- 12 ÷ 1 = 12 ✓ (1 and 12 are factors)
- 12 ÷ 2 = 6 ✓ (2 and 6 are factors)
- 12 ÷ 3 = 4 ✓ (3 and 4 are factors)
- 12 ÷ 4 = 3 ✓ (already counted)
- 12 ÷ 5 = 2 remainder 2 ✗
- 12 ÷ 6 = 2 ✓ (already counted)
- 12 ÷ 7 onwards: same pairs come back.
So the factors of 12 are: 1, 2, 3, 4, 6, 12 , six in all.
Every number has at least two factors: 1 and the number itself. A number with exactly two factors (just 1 and itself) is called a prime number. 2, 3, 5, 7, 11, 13 are prime.
Numbers with more than two factors are called composite numbers. 4, 6, 8, 9, 10, 12 are composite.
The number 1 is special , it has only one factor, itself. It is neither prime nor composite.
Tip: factors come in pairs. For 12: (1, 12), (2, 6), (3, 4). Once you cross the square root (about 3.5 for 12), the pairs start repeating.
Worked examples
Example 1. Find all factors of 18.
- 1 × 18, 2 × 9, 3 × 6.
- Factors: 1, 2, 3, 6, 9, 18.
Example 2. Is 23 a prime number?
- Try dividing by 2, 3, 4, 5… up to its square root (~4.8).
- None divides exactly.
- Yes, 23 is prime.
Example 3. Grandmother has 24 patches. List all rectangular arrangements.
- 1 × 24, 2 × 12, 3 × 8, 4 × 6.
- So factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
Try it
- List all factors of 15.
- List all factors of 20.
- Is 19 prime or composite?
- Find all rectangular arrangements of 36 patches.
- Which has more factors: 10 or 12?
Hands-on
Bead rectangles. Take 20 beads (or buttons). Try arranging them in rectangles. List all the rectangles you can make. The side numbers are the factors.
Prime hunt. List the first 20 numbers. Test each for being prime by trying to divide by 2, 3, 5, 7. Circle all the prime numbers. They are: 2, 3, 5, 7, 11, 13, 17, 19.