If A is the set of all integral multiples of 3 and B is the set of all integral multiples of 5, then `A nn B` is the set all intergal multiples of :

Options

A. `3+5`
B. `5-3`
C. GCD(3,5)
D. LCM(3,5)

Answer

To solve the problem, we need to find the intersection of the sets A and B, where:

- Set A consists of all integral multiples of 3.
- Set B consists of all integral multiples of 5.

### Step-by-Step Solution:

1. **Define the Sets:**
- Set A (multiples of 3): A = { ..., -6, -3, 0, 3, 6, 9, 12, 15, ... }
- Set B (multiples of 5): B = { ..., -10, -5, 0, 5, 10, 15, 20, 25, ... }

2. **Identify the Intersection:**
- The intersection of two sets, denoted as A ∩ B, includes all elements that are common to both sets.

3. **Find Common Multiples:**
- To find the common multiples of 3 and 5, we need to look for numbers that can be expressed as both 3n (for some integer n) and 5m (for some integer m).
- The smallest number that is a multiple of both 3 and 5 is their least common multiple (LCM).

4. **Calculate the LCM of 3 and 5:**
- The LCM of two numbers can be calculated using the formula:
\[ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} \]
- For 3 and 5:
- GCD(3, 5) = 1 (since 3 and 5 are coprime)
- Therefore,
\[ \text{LCM}(3, 5) = \frac{3 \times 5}{1} = 15 \]

5. **Conclusion:**
- The intersection A ∩ B consists of all integral multiples of 15.
- Therefore, A ∩ B = { ..., -30, -15, 0, 15, 30, 45, ... }

### Final Answer:
The set A ∩ B is the set of all integral multiples of **15**.

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