Grade 10 — Mathematics 10C › C10U04 Factors, Roots & Irrational Numbers

Grade 10 › C10U04 Factors, Roots & Irrational Numbers › L1
GRADE 10 · ALBERTAOUTCOME AB-MAT10C-AN1

L1 · Prime factorization

Learning goal

“I can write any whole number as a product of prime factors — its prime factorization — using a factor tree or a division ladder, and I can explain why 0 and 1 have no prime factors.”

Prime number — a whole number greater than 1 with exactly two distinct divisors: 1 and itself. 7 is prime; its only divisors are 1 and 7.
Composite number — a whole number greater than 1 that is not prime: it has a divisor other than 1 and itself. 12 is composite (divisors 1, 2, 3, 4, 6, 12).
Prime factor — a factor of a number that is itself prime. The prime factors of 12 are 2 and 3.
Prime factorization — a number written as a product of prime factors only, usually with exponents: 12 = 2² × 3. Every composite number has exactly one prime factorization.
Worked example

Find the prime factorization of 72. Step 1: Split 72 into any factor pair: 72 = 8 × 9. Step 2: Split anything composite: 8 = 2 × 4, and 4 = 2 × 2. 9 = 3 × 3. Step 3: Every branch now ends in a prime: 2, 2, 2, 3, 3. Stop. Step 4: Collect with exponents: 72 = 2³ × 3². Check by multiplying back: 2³ × 3² = 8 × 9 = 72 ✓ Say it plainly: it does not matter which factor pair you start with — every tree for 72 ends with the same primes. That is why the prime factorization is the factorization.

Check yourself — explain it

In the tree for 72, why did we split 4 but not 3?

Paired problem — example still visible

Find the prime factorization of 96. -> 96 = 2⁵ × 3 (96 = 8 × 12 = (2 × 2 × 2) × (2 × 2 × 3); five 2s and one 3. Check: 32 × 3 = 96 ✓)

Faded completion — last step is yours

"84 = 4 × 21 = (2 × ___) × (3 × ___), so 84 = 2^__ × 3 × 7" -> 2; 7; 2 (84 = 2² × 3 × 7)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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