Find the no. of zeros in the product of `(5 xx 10 xx 25 xx 40 xx 50 xx 55 xx 65 xx 125 xx 80)`

Options

A. 13
B. 9
C. 4
D. 5

Answer

To find the number of zeros in the product \(5 \times 10 \times 25 \times 40 \times 50 \times 55 \times 65 \times 125 \times 80\), we need to determine how many times the product can be divided by 10. Each 10 is made up of a factor of 2 and a factor of 5. Therefore, the number of zeros at the end of the product will be determined by the minimum number of pairs of 2s and 5s in the factorization of the product.

### Step-by-Step Solution:

1. **Factor Each Number**:
- \(5 = 5^1\)
- \(10 = 2^1 \times 5^1\)
- \(25 = 5^2\)
- \(40 = 2^3 \times 5^1\)
- \(50 = 2^1 \times 5^2\)
- \(55 = 5^1 \times 11^1\)
- \(65 = 5^1 \times 13^1\)
- \(125 = 5^3\)
- \(80 = 2^4 \times 5^0\)

2. **Count the Factors of 5**:
- From \(5\): \(1\)
- From \(10\): \(1\)
- From \(25\): \(2\)
- From \(40\): \(1\)
- From \(50\): \(2\)
- From \(55\): \(1\)
- From \(65\): \(1\)
- From \(125\): \(3\)
- From \(80\): \(0\)

**Total Factors of 5**:
\[ 1 + 1 + 2 + 1 + 2 + 1 + 1 + 3 + 0 = 12 \]

3. **Count the Factors of 2**:
- From \(5\): \(0\)
- From \(10\): \(1\)
- From \(25\): \(0\)
- From \(40\): \(3\)
- From \(50\): \(1\)
- From \(55\): \(0\)
- From \(65\): \(0\)
- From \(125\): \(0\)
- From \(80\): \(4\)

**Total Factors of 2**:
\[ 0 + 1 + 0 + 3 + 1 + 0 + 0 + 0 + 4 = 9 \]

4. **Determine the Number of Pairs of 2s and 5s**:
The number of pairs of \(2\) and \(5\) is the minimum of the total counts:
\[ \text{Pairs} = \min(12, 9) = 9 \]

5. **Add the Zeros from the Factors of 10**:
Since we also have \(4\) zeros from the \(10\) in the product, we add this to the pairs we found:
\[ \text{Total Zeros} = 9 + 4 = 13 \]

### Final Answer:
The total number of zeros in the product \(5 \times 10 \times 25 \times 40 \times 50 \times 55 \times 65 \times 125 \times 80\) is **13**.