1. **Problem statement:** Given the numbers 300, 144, and 108, we need to: a. Create factor trees for each. b. Express each as a product of prime factors. c. Find their greatest common factor (GCF). d. Find their least common multiple (LCM). 2. **Prime factorization:** To find prime factors, we repeatedly divide by the smallest prime until we reach 1. 3. **Factor trees and prime factorizations:** - 300: Divide by 2: $$300 = 2 \times 150$$ Divide 150 by 2: $$150 = 2 \times 75$$ Divide 75 by 3: $$75 = 3 \times 25$$ Divide 25 by 5: $$25 = 5 \times 5$$ So, $$300 = 2 \times 2 \times 3 \times 5 \times 5 = 2^2 \times 3 \times 5^2$$ - 144: Divide by 2: $$144 = 2 \times 72$$ Divide 72 by 2: $$72 = 2 \times 36$$ Divide 36 by 2: $$36 = 2 \times 18$$ Divide 18 by 2: $$18 = 2 \times 9$$ Divide 9 by 3: $$9 = 3 \times 3$$ So, $$144 = 2^4 \times 3^2$$ - 108: Divide by 2: $$108 = 2 \times 54$$ Divide 54 by 2: $$54 = 2 \times 27$$ Divide 27 by 3: $$27 = 3 \times 9$$ Divide 9 by 3: $$9 = 3 \times 3$$ So, $$108 = 2^2 \times 3^3$$ 4. **Greatest Common Factor (GCF):** Take the minimum powers of common primes: - For 2: min(2,4,2) = 2 - For 3: min(1,2,3) = 1 - For 5: min(2,0,0) = 0 (5 is not common to all) So, $$\text{GCF} = 2^2 \times 3^1 = 4 \times 3 = 12$$ 5. **Least Common Multiple (LCM):** Take the maximum powers of all primes: - For 2: max(2,4,2) = 4 - For 3: max(1,2,3) = 3 - For 5: max(2,0,0) = 2 So, $$\text{LCM} = 2^4 \times 3^3 \times 5^2 = 16 \times 27 \times 25$$ Calculate stepwise: $$16 \times 27 = 432$$ $$432 \times 25 = 10800$$ **Final answers:** - Prime factorizations: - 300 = $2^2 \times 3 \times 5^2$ - 144 = $2^4 \times 3^2$ - 108 = $2^2 \times 3^3$ - GCF = 12 - LCM = 10800