If A={x:x is multiple of 2). B ={x: x is a multiple of 5} and C={x: x is a multiple of 10} then `A cap (B cap C)` is equal to

Options

A. A
B. B
C. C
D. {x: x is a multiple of 100}

Answer

To solve the problem, we need to find the intersection of the sets A, B, and C defined as follows:

1. **Set A**: A = {x : x is a multiple of 2}
- This means A includes numbers like 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on.

2. **Set B**: B = {x : x is a multiple of 5}
- This means B includes numbers like 5, 10, 15, 20, 25, 30, and so on.

3. **Set C**: C = {x : x is a multiple of 10}
- This means C includes numbers like 10, 20, 30, 40, 50, and so on.

Now, we need to find the intersection of these sets, specifically \( A \cap (B \cap C) \).

### Step-by-Step Solution:

**Step 1: Find B ∩ C**
- The intersection of sets B and C includes all elements that are common to both sets.
- Set B includes multiples of 5: {5, 10, 15, 20, 25, 30, ...}
- Set C includes multiples of 10: {10, 20, 30, 40, ...}
- The common elements (intersection) are the multiples of 10, which are also multiples of 5.
- Therefore, \( B \cap C = C = \{10, 20, 30, 40, ...\} \).

**Step 2: Find A ∩ (B ∩ C)**
- Now we need to find the intersection of set A with the result from Step 1.
- Set A includes multiples of 2: {2, 4, 6, 8, 10, 12, 14, ...}
- Set C (which is \( B \cap C \)) includes multiples of 10: {10, 20, 30, 40, ...}
- The common elements (intersection) between A and C are the multiples of 10, since every multiple of 10 is also a multiple of 2.
- Therefore, \( A \cap (B \cap C) = C = \{10, 20, 30, 40, ...\} \).

### Conclusion:
Thus, \( A \cap (B \cap C) = C \), which means the answer is set C.

### Final Answer:
**A ∩ (B ∩ C) = C = {x : x is a multiple of 10}**

---