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

Grade 10 › C10U04 Factors, Roots & Irrational Numbers › L2
GRADE 10 · ALBERTAOUTCOME AB-MAT10C-AN1

L2 · Greatest common factor and lowest common multiple

Learning goal

“I can find the greatest common factor (GCF) and the least common multiple (LCM) of a set of whole numbers using prime factorizations, and I can explain the process — and use them to solve packing and scheduling problems.”

Multiple — the result of multiplying a number by a whole number: the multiples of 8 are 8, 16, 24, 32, …
Common multiple — a multiple shared by two or more numbers: 24 is a common multiple of 8 and 12.
Greatest common factor (GCF) — the largest number that divides every number in a set. GCF(12, 18) = 6.
Least common multiple (LCM) — the smallest positive number that is a multiple of every number in a set. LCM(8, 12) = 24.
Worked example

Find GCF(48, 180). Step 1: Factor both: 48 = 2⁴ × 3 and 180 = 2² × 3² × 5. Step 2: A common factor can only use primes that appear in BOTH factorizations: 2 and 3 (the 5 in 180 has no partner in 48). Step 3: For each shared prime, take the SMALLER exponent — the factor must divide both numbers, so it cannot contain more 2s than the stingier number has: 2² (from 180) and 3¹ (from 48). Step 4: GCF = 2² × 3 = 12. Check: 48 ÷ 12 = 4 and 180 ÷ 12 = 15, both whole ✓ Say it plainly: GCF = shared primes, smallest exponents.

Check yourself — explain it

Why does the GCF of 48 and 180 use 2² and not 2⁴?

Paired problem — example still visible

Find GCF(84, 126). -> 42 (84 = 2² × 3 × 7; 126 = 2 × 3² × 7; shared at smaller exponents: 2 × 3 × 7 = 42. Check: 84 ÷ 42 = 2, 126 ÷ 42 = 3 ✓)

Faded completion — last step is yours

"72 = 2³ × 3² and 120 = 2³ × 3 × 5 share 2³ and 3, so GCF(72, 120) = 2³ × ___ = ___" -> 3; 24

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