Find the number of Zeros at the end of the product -
`12 xx 5 xx 15 xx 24 xx 13 xx 30 xx 75 `

Options

A. 4
B. 5
C. 2
D. 3

Answer

To find the number of zeros at the end of the product \( 12 \times 5 \times 15 \times 24 \times 13 \times 30 \times 75 \), we need to determine how many times the factors 10 can be formed in the product. A factor of 10 is formed by multiplying 2 and 5 together. Therefore, we need to count the number of 2s and 5s in the prime factorization of the numbers in the product.

### Step 1: Prime Factorization of Each Number
Let's factor each number in the product:

1. **12**:
\[ 12 = 4 \times 3 = 2^2 \times 3^1 \]
(Number of 2s = 2)

2. **5**:
\[ 5 = 5^1 \]
(Number of 5s = 1)

3. **15**:
\[ 15 = 3 \times 5 = 5^1 \times 3^1 \]
(Number of 5s = 1)

4. **24**:
\[ 24 = 8 \times 3 = 2^3 \times 3^1 \]
(Number of 2s = 3)

5. **13**:
\[ 13 = 13^1 \]
(No 2s or 5s)

6. **30**:
\[ 30 = 10 \times 3 = 2^1 \times 5^1 \times 3^1 \]
(Number of 2s = 1, Number of 5s = 1)

7. **75**:
\[ 75 = 25 \times 3 = 5^2 \times 3^1 \]
(Number of 5s = 2)

### Step 2: Count the Total Number of 2s and 5s
Now, we will sum the total number of 2s and 5s from all the factors:

- Total number of 2s:
- From 12: 2
- From 5: 0
- From 15: 0
- From 24: 3
- From 13: 0
- From 30: 1
- From 75: 0

Total number of 2s = \( 2 + 0 + 0 + 3 + 0 + 1 + 0 = 6 \)

- Total number of 5s:
- From 12: 0
- From 5: 1
- From 15: 1
- From 24: 0
- From 13: 0
- From 30: 1
- From 75: 2

Total number of 5s = \( 0 + 1 + 1 + 0 + 0 + 1 + 2 = 5 \)

### Step 3: Determine the Number of Pairs of (2, 5)
To find the number of zeros at the end of the product, we need to find the minimum of the total number of 2s and 5s:

\[ \text{Number of pairs} = \min(\text{Total number of 2s}, \text{Total number of 5s}) = \min(6, 5) = 5 \]

### Conclusion
The number of zeros at the end of the product \( 12 \times 5 \times 15 \times 24 \times 13 \times 30 \times 75 \) is **5**.

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