Common Mistakes with Factors, Multiples and Primes
These are the errors that appear most often in tests and homework. Knowing them in advance is the best defence against making them yourself.
Mistake 1 – Confusing Factors and Multiples
Writing “multiples of 6 are 1, 2, 3, 6” — those are factors, not multiples!
Mistake 2 – Thinking 1 Is Prime
1 is not prime. By definition, a prime has exactly two factors. 1 has only one factor (itself), so it fails the test.
Mistake 3 – Missing Factors When Listing
Always work in pairs and check up to the square root. 36 = √36 = 6. Test 1–6: (1,36), (2,18), (3,12), (4,9), (6,6).
Mistake 4 – Taking Highest Power for GCF
GCF uses the LOWEST power of each shared prime. LCM uses the HIGHEST. Swapping these is a very common error.
| Calculation | Rule | Correct Example (12=2²×3, 18=2×3²) |
|---|---|---|
| GCF | Lowest shared powers | 2¹ × 3¹ = 6 |
| LCM | Highest powers | 2² × 3² = 36 |
Mistake 5 – Applying GCF × LCM = a × b to Three Numbers
This formula applies ONLY to two numbers. For three numbers, find prime factorizations instead.
Mistake 6 – Stopping the Euclidean Algorithm Too Early
Stop only when the remainder is exactly 0, not when it is 1.
35 ÷ 14 = 2 r 7. Then 14 ÷ 7 = 2 r 0. GCF = 7. (Some students stop at remainder 7 and call GCF 7 incorrectly for the wrong reason — always carry through until remainder = 0.)
Mistake 7 – Incorrect Prime Factorization
Forgetting that a branch must end at a prime. Composite tips must still be split.
Key Takeaways
- Factors: finite, divide IN. Multiples: infinite, the number goes into them.
- 1 is neither prime nor composite.
- GCF = lowest shared prime powers; LCM = highest prime powers.
- GCF × LCM = a × b applies to two numbers only.
- In the Euclidean algorithm, stop only at remainder 0.
Quick Self-Check
- Is 1 prime? Why or why not?
- Factors or multiples of 8: {8, 16, 24, 32}?
- For 8=2³ and 12=2²×3: what is the GCF and the LCM?
- What is GCF(10, 15, 21)?
- In a factor tree for 60, which of these is an error: final tip of 9? of 2? of 7?