QuestionDecember 25, 2025

5. Find the GCF and LCM of the numbers given below. a. 36,60 b. 84,224 c. 15,39,105 d. 16,20,48

5. Find the GCF and LCM of the numbers given below. a. 36,60 b. 84,224 c. 15,39,105 d. 16,20,48
5. Find the GCF and LCM of the numbers given below. a. 36,60 b. 84,224 c. 15,39,105 d. 16,20,48

Solution
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Answer

a. GCF: 12, LCM: 180 ### b. GCF: 28, LCM: 672 ### c. GCF: 3, LCM: 1365 ### d. GCF: 4, LCM: 240 Explanation 1. Prime factorization 36 = 2^2 \times 3^2, 60 = 2^2 \times 3 \times 5; 84 = 2^2 \times 3 \times 7, 224 = 2^5 \times 7; 15 = 3 \times 5, 39 = 3 \times 13, 105 = 3 \times 5 \times 7; 16 = 2^4, 20 = 2^2 \times 5, 48 = 2^4 \times 3. 2. Find GCF (Greatest Common Factor) Use \text{GCF} = product of lowest powers of common primes. - a. 2^2 \times 3 = 12 - b. 2^2 \times 7 = 28 - c. 3 (only common prime) - d. 2^2 = 4 3. Find LCM (Least Common Multiple) Use \text{LCM} = product of highest powers of all primes present. - a. 2^2 \times 3^2 \times 5 = 180 - b. 2^5 \times 3 \times 7 = 672 - c. 3 \times 5 \times 7 \times 13 = 1365 - d. 2^4 \times 3 \times 5 = 240

Explanation

1. Prime factorization<br /> 36 = $2^2 \times 3^2$, 60 = $2^2 \times 3 \times 5$; 84 = $2^2 \times 3 \times 7$, 224 = $2^5 \times 7$; 15 = $3 \times 5$, 39 = $3 \times 13$, 105 = $3 \times 5 \times 7$; 16 = $2^4$, 20 = $2^2 \times 5$, 48 = $2^4 \times 3$.<br />2. Find GCF (Greatest Common Factor)<br /> Use $\text{GCF} = $ product of lowest powers of common primes.<br />- a. $2^2 \times 3 = 12$<br />- b. $2^2 \times 7 = 28$<br />- c. $3$ (only common prime)<br />- d. $2^2 = 4$<br />3. Find LCM (Least Common Multiple)<br /> Use $\text{LCM} = $ product of highest powers of all primes present.<br />- a. $2^2 \times 3^2 \times 5 = 180$<br />- b. $2^5 \times 3 \times 7 = 672$<br />- c. $3 \times 5 \times 7 \times 13 = 1365$<br />- d. $2^4 \times 3 \times 5 = 240$
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