Factors and Multiples • Topic 2 of 7

Composite Numbers

A composite number is a whole number greater than 1 that has more than two factors — that is, it can be divided evenly by some number other than 1 and itself. Examples are 4, 6, 8, 9, 10, 12, … . Every whole number above 1 is either prime or composite; 1 is neither. A useful idea is the semiprime: a composite that is the product of exactly two primes (such as 6 = 2 × 3 or 9 = 3 × 3). The number of factors a composite has can be found from its prime factorization, a frequent SAT shortcut.

✅ Solved examples

1. List all composite numbers between 10 and 20.
Check each: 11, 13, 17, 19 are prime. The rest have extra factors: 12, 14, 15, 16, 18 (and we exclude the primes). So the composites are 12, 14, 15, 16, 18.
2. How many factors does the composite number 12 have?
Factor pairs of 12: 1×12, 2×6, 3×4. Listing all divisors gives 1, 2, 3, 4, 6, 12 — that is 6 factors. (Check via 12 = 2² × 3: number of factors = (2+1)(1+1) = 6.)
3. Which is the smallest composite number?
Start above 1. The number 2 and 3 are prime. The next whole number, 4 = 2 × 2, has factors 1, 2, 4 — three factors — so it is composite. The smallest composite number is 4.
4. Is 1 composite?
No. A composite number must have more than two factors. The number 1 has exactly one factor (itself), so it is neither prime nor composite. This special case is often tested.

✏️ Practice — try these, take hints as needed

1. How many composite numbers are there between 1 and 10?
1 is neither prime nor composite; ignore it.
Remove the primes 2, 3, 5, 7 from 2–10.
What remains — 4, 6, 8, 9, 10 — are composite.
5 (namely 4, 6, 8, 9, 10).
2. Which number is composite: 29, 31, 33, 37?
Three of these are prime; find the odd one out.
Test divisibility by 3 using digit sums.
33 has digit sum 6 (÷3), so 33 = 3 × 11.
33.
3. Find the smallest composite number greater than 30.
Start at 31 and move up.
31 is prime; check 32.
32 = 2⁵, which has many factors.
32.
4. A composite number equals the product of exactly two primes and is even and less than 10. What is it?
Even means one prime factor is 2.
The number = 2 × (a prime), and is under 10.
Options: 2×2=4, 2×3=6 — both qualify; list them.
4 (=2×2) and 6 (=2×3).
5. How many factors does 16 have, and is it composite?
Write 16 as a power of 2.
16 = 2⁴, so number of factors = 4 + 1.
More than two factors means composite.
5 factors (1,2,4,8,16); yes, composite.

📝 Topic test — 8 questions

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