How many zeroes are there in product `4^(6) xx 15^(10)`?

Options

A. a)10
B. b)12
C. c)11
D. d)6

Answer

To find the number of zeroes in the product \( 4^6 \times 15^{10} \), we need to determine how many pairs of the factors 2 and 5 can be formed, since each pair contributes to one trailing zero in the final product.

### Step-by-Step Solution:

1. **Express the numbers in terms of their prime factors**:
- \( 4 = 2^2 \), so \( 4^6 = (2^2)^6 = 2^{12} \).
- \( 15 = 3 \times 5 \), so \( 15^{10} = (3 \times 5)^{10} = 3^{10} \times 5^{10} \).

2. **Combine the factors**:
- Now, we can express the entire product:
\[ 4^6 \times 15^{10} = 2^{12} \times (3^{10} \times 5^{10}) = 2^{12} \times 3^{10} \times 5^{10}. \]

3. **Identify the number of factors of 2 and 5**:
- From the expression \( 2^{12} \times 3^{10} \times 5^{10} \):
- The number of factors of 2 is 12.
- The number of factors of 5 is 10.

4. **Determine the number of pairs of 2 and 5**:
- Each pair of one 2 and one 5 contributes to one trailing zero.
- The number of pairs is determined by the smaller count of the two factors:
\[ \text{Number of pairs} = \min(12, 10) = 10. \]

5. **Conclusion**:
- Therefore, the number of trailing zeroes in the product \( 4^6 \times 15^{10} \) is **10**.

### Final Answer:
The number of zeroes in the product \( 4^6 \times 15^{10} \) is **10**.