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) $$ \operatorname{GCD}(12,16,8)=\square $$

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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**: $$ 840 
eq 5040 $$
- (a) $$ \operatorname{GCD}(12, 16, 8) = 4 $$ - (b) $$ \operatorname{GCD}(12, 30, 14) = 2 $$ and $$ \operatorname{LCM}(12, 30, 14) = 420 $$ - (c) It is **not true** that $$ \operatorname{GCD}(12, 30, 14) \cdot \operatorname{LCM}(12, 30, 14) = 12 \cdot 30 \cdot 14 $$.

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