Use the method of intersection of sets to compute the following. (a) \( \operatorname{GCD}(12,16,8) \) and \( \operatorname{LCM}(12,16,8) \) (b) \( \operatorname{GCD}(12,30,14) \) and \( \operatorname{LCM}(12,30,14) \) (c) Is it true that \( \operatorname{GCD}(12,30,14) \cdot \operatorname{LCM}(12,30,14)=12 \cdot 30 \cdot 14 \) ? (a) \( \operatorname{GCD}(12,16,8)=\square \)

To compute the GCD and LCM using the method of intersection of sets, we first need to find the prime factorization of each number involved in the calculations. ### (a) \( \operatorname{GCD}(12, 16, 8) \) and \( \operatorname{LCM}(12, 16, 8) \) 1. **Prime Factorization**: - \( 12 = 2^2 \cdot 3^1 \) - \( 16 = 2^4 \) - \( 8 = 2^3 \) 2. **Finding GCD**: - The GCD is found by taking the lowest power of each prime that appears in all factorizations. - For \( 2 \): The minimum power is \( 2^2 \) (from 12). - For \( 3 \): It does not appear in 16 and 8, so we do not include it. - Thus, \( \operatorname{GCD}(12, 16, 8) = 2^2 = 4 \). 3. **Finding LCM**: - The LCM is found by taking the highest power of each prime that appears in any of the factorizations. - For \( 2 \): The maximum power is \( 2^4 \) (from 16). - For \( 3 \): The maximum power is \( 3^1 \) (from 12). - Thus, \( \operatorname{LCM}(12, 16, 8) = 2^4 \cdot 3^1 = 16 \cdot 3 = 48 \). ### Summary for (a): - \( \operatorname{GCD}(12, 16, 8) = 4 \) - \( \operatorname{LCM}(12, 16, 8) = 48 \) ### (b) \( \operatorname{GCD}(12, 30, 14) \) and \( \operatorname{LCM}(12, 30, 14) \) 1. **Prime Factorization**: - \( 12 = 2^2 \cdot 3^1 \) - \( 30 = 2^1 \cdot 3^1 \cdot 5^1 \) - \( 14 = 2^1 \cdot 7^1 \) 2. **Finding GCD**: - For \( 2 \): The minimum power is \( 2^1 \) (from 30 and 14). - For \( 3 \): The minimum power is \( 3^0 \) (not present in 14). - For \( 5 \): It does not appear in 12 and 14, so we do not include it. - For \( 7 \): It does not appear in 12 and 30, so we do not include it. - Thus, \( \operatorname{GCD}(12, 30, 14) = 2^1 = 2 \). 3. **Finding LCM**: - For \( 2 \): The maximum power is \( 2^2 \) (from 12). - For \( 3 \): The maximum power is \( 3^1 \) (from 12 and 30). - For \( 5 \): The maximum power is \( 5^1 \) (from 30). - For \( 7 \): The maximum power is \( 7^1 \) (from 14). - Thus, \( \operatorname{LCM}(12, 30, 14) = 2^2 \cdot 3^1 \cdot 5^1 \cdot 7^1 = 4 \cdot 3 \cdot 5 \cdot 7 = 420 \). ### Summary for (b): - \( \operatorname{GCD}(12, 30, 14) = 2 \) - \( \operatorname{LCM}(12, 30, 14) = 420 \) ### (c) Is it true that \( \operatorname{GCD}(12, 30, 14) \cdot \operatorname{LCM}(12, 30, 14) = 12 \cdot 30 \cdot 14 \)? 1. **Calculating the left side**: - \( \operatorname{GCD}(12, 30, 14) \cdot \operatorname{LCM}(12, 30, 14) = 2 \cdot 420 = 840 \). 2. **Calculating the right side**: - \( 12 \cdot 30 \cdot 14 = 360 \cdot 14 = 5040 \). 3. **Comparison**: - Left side: \( 840 \) - Right side: \( 5040 \) ### Conclusion for (c): - It is **not true** that \( \operatorname{GCD}(12, 30, 14) \cdot \operatorname{LCM}(12, 30, 14) = 12 \cdot 30 \cdot 14 \). ### Final Answers: - (a) \( \operatorname{GCD}(12, 16, 8) = 4 \) - (b) \( \operatorname{GCD}(12, 30, 14) = 2 \) and \( \operatorname{LCM}(12, 30, 14) = 420 \) - (c) **False**: \( 840 \neq 5040 \)