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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**(a) Calculation of $$\operatorname{GCD}(12,16,8)$$ and $$\operatorname{LCM}(12,16,8)$$:**

1. **Find the prime factorization:**

$$
   12 = 2^2 \cdot 3,\quad 16 = 2^4,\quad 8 = 2^3
   $$

2. **Determine the common prime factors and their minimum exponents for the GCD:**

- The only common prime factor is $$2$$.
   - Minimum exponent of $$2$$ among the numbers is $$\min(2,4,3) = 2$$.

Thus,

$$
   \operatorname{GCD}(12,16,8) = 2^2 = 4.
   $$

3. **Determine the primes with their maximum exponents for the LCM:**

- For $$2$$: maximum exponent is $$\max(2,4,3) = 4$$.
   - The number $$12$$ contributes the prime $$3$$ (with exponent $$1$$).

Thus,

$$
   \operatorname{LCM}(12,16,8) = 2^4 \cdot 3 = 16 \cdot 3 = 48.
   $$

4. **Answer for part (a):**

$$
   \operatorname{GCD}(12,16,8)=\boxed{4},\quad \operatorname{LCM}(12,16,8)=48.
   $$

---

**(b) Calculation of $$\operatorname{GCD}(12,30,14)$$ and $$\operatorname{LCM}(12,30,14)$$:**

1. **Find the prime factorization:**

$$
   12 = 2^2 \cdot 3,\quad 30 = 2 \cdot 3 \cdot 5,\quad 14 = 2 \cdot 7.
   $$

2. **Determine the common prime factors and their minimum exponents for the GCD:**

- The only common prime factor is $$2$$.
   - Minimum exponent of $$2$$ among the numbers is $$\min(2,1,1) = 1$$.

Thus,

$$
   \operatorname{GCD}(12,30,14) = 2^1 = 2.
   $$

3. **Determine the primes with their maximum exponents for the LCM:**

- For $$2$$: maximum exponent is $$2$$ (from $$12$$).
   - Include $$3$$ (from $$12$$ and $$30$$), exponent $$1$$.
   - Include $$5$$ (from $$30$$), exponent $$1$$.
   - Include $$7$$ (from $$14$$), exponent $$1$$.

Thus,

$$
   \operatorname{LCM}(12,30,14) = 2^2 \cdot 3 \cdot 5 \cdot 7 = 4 \cdot 3 \cdot 5 \cdot 7 = 420.
   $$

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**(c) Verification of the relationship**

Check if

$$
\operatorname{GCD}(12,30,14) \cdot \operatorname{LCM}(12,30,14) = 12 \cdot 30 \cdot 14.
$$

Using our computed values:

- $$\operatorname{GCD}(12,30,14) = 2$$
- $$\operatorname{LCM}(12,30,14) = 420$$

Then:

$$
2 \cdot 420 = 840.
$$

Now, compute:

$$
12 \cdot 30 \cdot 14.
$$

First, compute $$12 \cdot 30 = 360$$, then:

$$
360 \cdot 14 = 5040.
$$

Since $$840 \neq 5040$$, the equality

$$
\operatorname{GCD}(12,30,14) \cdot \operatorname{LCM}(12,30,14)=12 \cdot 30 \cdot 14
$$

is not true.

---

**Summary of Answers:**

- (a) $$\operatorname{GCD}(12,16,8)=\boxed{4}$$ and $$\operatorname{LCM}(12,16,8)=48$$.
- (b) $$\operatorname{GCD}(12,30,14)=2$$ and $$\operatorname{LCM}(12,30,14)=420$$.
- (c) The equality $$\operatorname{GCD}(12,30,14) \cdot \operatorname{LCM}(12,30,14)=12 \cdot 30 \cdot 14$$ is false.
**(a) $$\operatorname{GCD}(12,16,8)=4$$ and $$\operatorname{LCM}(12,16,8)=48$$.** **(b) $$\operatorname{GCD}(12,30,14)=2$$ and $$\operatorname{LCM}(12,30,14)=420$$.** **(c) The equality $$\operatorname{GCD}(12,30,14) \cdot \operatorname{LCM}(12,30,14)=12 \cdot 30 \cdot 14$$ is not true.

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