How many zeroes are there at the end of product `35 xx 27 xx 52`?

Options

A. 1
B. 6
C. 2
D. 4

Answer

To find how many zeroes are at the end of the product \(35 \times 27 \times 52\), we need to determine how many times the product can be divided by 10. Since \(10 = 2 \times 5\), we need to find the number of pairs of factors 2 and 5 in the product.

### Step 1: Factor each number
Let's factor each number in the product:

1. **Factor 35**:
\[ 35 = 5 \times 7 \]

2. **Factor 27**:
\[ 27 = 3^3 \]

3. **Factor 52**:
\[ 52 = 2^2 \times 13 \]

### Step 2: Combine the factors
Now, we can combine all the factors:
\[ 35 \times 27 \times 52 = (5 \times 7) \times (3^3) \times (2^2 \times 13) \]

### Step 3: Count the factors of 2 and 5
Next, we count how many times 2 and 5 appear in the factorization:

- From \(35\), we have **1 factor of 5**.
- From \(27\), we have **0 factors of 2 or 5**.
- From \(52\), we have **2 factors of 2**.

So, in total:
- **Total factors of 2**: \(2\) (from 52)
- **Total factors of 5**: \(1\) (from 35)

### Step 4: Determine the number of pairs of (2, 5)
To find the number of zeroes at the end of the product, we take the minimum of the counts of 2 and 5:
\[ \text{Number of zeroes} = \min(\text{number of 2s}, \text{number of 5s}) = \min(2, 1) = 1 \]

### Conclusion
Thus, there is **1 zero** at the end of the product \(35 \times 27 \times 52\).