Find the number of Zeros at the end of the product -
`2^1 xx 5^2 xx 2^3 xx 5^4 xx2^5 xx 5^6 xx 2^7 xx 5^8 xx 2^9 xx 5^10 `

Options

A. 30
B. 25
C. 55
D. 50

Answer

To find the number of zeros at the end of the product \(2^1 \times 5^2 \times 2^3 \times 5^4 \times 2^5 \times 5^6 \times 2^7 \times 5^8 \times 2^9 \times 5^{10}\), we need to determine how many pairs of the factors 2 and 5 can be formed, since each pair contributes to a trailing zero.

### Step-by-Step Solution:

1. **Group the terms by their bases**:
The expression can be rewritten by grouping the powers of 2 and 5:
\[ (2^1 \times 2^3 \times 2^5 \times 2^7 \times 2^9) \times (5^2 \times 5^4 \times 5^6 \times 5^8 \times 5^{10}) \]

2. **Combine the powers of 2**:
To combine the powers of 2, we add the exponents:
\[ 1 + 3 + 5 + 7 + 9 \]
Calculating this:
- \(1 + 3 = 4\)
- \(4 + 5 = 9\)
- \(9 + 7 = 16\)
- \(16 + 9 = 25\)

Therefore, the total power of 2 is \(25\).

3. **Combine the powers of 5**:
Similarly, we add the exponents of 5:
\[ 2 + 4 + 6 + 8 + 10 \]
Calculating this:
- \(2 + 4 = 6\)
- \(6 + 6 = 12\)
- \(12 + 8 = 20\)
- \(20 + 10 = 30\)

Therefore, the total power of 5 is \(30\).

4. **Determine the number of pairs of 2 and 5**:
To find the number of trailing zeros, we need to find the minimum of the total powers of 2 and 5:
\[ \text{Pairs} = \min(25, 30) = 25 \]

5. **Conclusion**:
The number of trailing zeros in the product is equal to the number of pairs of 2 and 5, which is \(25\).

### Final Answer:
The number of zeros at the end of the product is **25**.