Factors, Multiples and Prime Factorization
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Learning Objectives
- 1List every factor of a whole number from 1 up to and including the number itself
- 2Classify whole numbers as prime or composite by counting their factors
- 3Build a factor tree and write the prime factorization using exponents
- 4Generate the first several multiples of a whole number by skip counting
Mini Lesson
Whole numbers can be broken apart in two useful ways. Factors are the smaller numbers that divide a whole number evenly with no remainder. Multiples are what you get when you skip-count the number forward. Every counting number greater than 1 is either a prime — built from no factors except 1 and itself — or a composite, which can always be rewritten as a product of primes.
Factors — the Numbers That Divide Evenly
- A factor of a whole number divides into it with NO remainder.
- Every whole number has 1 and itself as factors — these are the "boundary" factors.
- Find factors in pairs: for 12, ask "what times what equals 12?" — 1 x 12, 2 x 6, 3 x 4.
- Factors of 12: 1, 2, 3, 4, 6, 12. List them from least to greatest.
- Factors of 18: 1, 2, 3, 6, 9, 18.
Multiples — the Skip-Counting Numbers
- A multiple of a whole number is any product of that number times a counting number (1, 2, 3, ...).
- The first five multiples of 6 are 6, 12, 18, 24, 30 — the answers to 6 x 1, 6 x 2, 6 x 3, 6 x 4, 6 x 5.
- Every whole number is a multiple of itself (n x 1 = n) and a multiple of 1 (1 x n = n).
- Multiples never end — you can keep skip-counting forever.
Prime vs Composite
- A prime number has EXACTLY two distinct factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23.
- A composite number has MORE than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18.
- The number 1 is neither prime nor composite — it has only one factor.
- The number 2 is the ONLY even prime. Every other even number is composite (it has 2 as a factor).
Prime Factorization with a Factor Tree
- Prime factorization writes a composite number as a product of primes only.
- Build a factor tree: start with the number, branch into any factor pair, then keep branching composites until every leaf is prime.
- Example — factor tree of 24: 24 → 4 x 6 → (2 x 2) x (2 x 3). Primes at the leaves: 2, 2, 2, 3.
- Write the answer with exponents to group repeats: 24 = 2 x 2 x 2 x 3 = 2³ x 3.
- Example — factor tree of 36: 36 → 4 x 9 → (2 x 2) x (3 x 3). So 36 = 2² x 3².
Tip: when you finish a factor tree, the primes at the bottom can be written in any order — but always list them from smallest to largest with exponents (e.g. 2³ x 3, not 3 x 2³). Every composite has exactly one prime factorization.
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