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.

Most of the mistakes below share a common root: mixing up two similar-sounding rules (factors vs multiples, lowest vs highest prime power, when to stop a division). None of them require new knowledge to fix — just a moment’s pause to check which rule actually applies before writing an answer down. Working through the checklist here once, deliberately, tends to prevent the same slip from happening again under exam pressure.

Mistake 1 – Confusing Factors and Multiples

Error

Writing “multiples of 6 are 1, 2, 3, 6” — those are factors, not multiples!

Fix: Factors divide INTO the number (1, 2, 3, 6 divide into 6). Multiples are the number multiplied by 1, 2, 3... (6, 12, 18...)

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

Error: factors of 36 listed as 1, 2, 3, 6, 9, 12, 36 — missing 4 and 18

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.

GCF(35, 14) — common error

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.

Error: 36 = 6 × 6 (left as is). Fix: 6 = 2 × 3, so 36 = 2² × 3².

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: Spot the Mistake

Find the correct answer.