The number of zeros at the end of the product of :
`2^3xx3^4xx4^5xx5^6 + 3^5xx5^7xx7^9xx8^(10) + 4^5xx5^6xx6^7xx7^8-10^2xx15^3xx20^4` is :

Options

A. 5
B. 6
C. 28
D. none of these

Answer

To find the number of zeros at the end of the expression given, we need to analyze each term in the expression and determine how many pairs of factors of 2 and 5 can be formed, as each pair contributes one zero to the final product.

The expression is:
\[ 2^3 \times 3^4 \times 4^5 \times 5^6 + 3^5 \times 5^7 \times 7^9 \times 8^{10} + 4^5 \times 5^6 \times 6^7 \times 7^8 - 10^2 \times 15^3 \times 20^4 \]

### Step 1: Analyze the first term
**Term 1:** \( 2^3 \times 3^4 \times 4^5 \times 5^6 \)

- Rewrite \( 4^5 \) as \( (2^2)^5 = 2^{10} \).
- Total number of 2's: \( 2^3 + 2^{10} = 2^{13} \).
- Total number of 5's: \( 5^6 \).

Now, we can form pairs of (2, 5):
- Pairs of (2, 5) = \( \min(13, 6) = 6 \).

### Step 2: Analyze the second term
**Term 2:** \( 3^5 \times 5^7 \times 7^9 \times 8^{10} \)

- Rewrite \( 8^{10} \) as \( (2^3)^{10} = 2^{30} \).
- Total number of 2's: \( 2^{30} \).
- Total number of 5's: \( 5^7 \).

Now, we can form pairs of (2, 5):
- Pairs of (2, 5) = \( \min(30, 7) = 7 \).

### Step 3: Analyze the third term
**Term 3:** \( 4^5 \times 5^6 \times 6^7 \times 7^8 \)

- Rewrite \( 4^5 \) as \( 2^{10} \) and \( 6^7 \) as \( (2 \times 3)^7 = 2^7 \times 3^7 \).
- Total number of 2's: \( 2^{10} + 2^7 = 2^{17} \).
- Total number of 5's: \( 5^6 \).

Now, we can form pairs of (2, 5):
- Pairs of (2, 5) = \( \min(17, 6) = 6 \).

### Step 4: Analyze the fourth term
**Term 4:** \( 10^2 \times 15^3 \times 20^4 \)

- Rewrite \( 10^2 \) as \( (2 \times 5)^2 = 2^2 \times 5^2 \).
- Rewrite \( 15^3 \) as \( (3 \times 5)^3 = 3^3 \times 5^3 \).
- Rewrite \( 20^4 \) as \( (2^2 \times 5)^4 = 2^8 \times 5^4 \).

Now, total number of 2's and 5's:
- Total number of 2's: \( 2^2 + 2^8 = 2^{10} \).
- Total number of 5's: \( 5^2 + 5^3 + 5^4 = 2 + 3 + 4 = 9 \).

Now, we can form pairs of (2, 5):
- Pairs of (2, 5) = \( \min(10, 9) = 9 \).

### Step 5: Combine results
Now we have the number of zeros from each term:
- Term 1: 6 zeros
- Term 2: 7 zeros
- Term 3: 6 zeros
- Term 4: 9 zeros

### Step 6: Determine the final count of zeros
When we add or subtract terms, the total number of zeros in the final result will be determined by the term with the minimum number of zeros:
- Minimum from (6, 7, 6, 9) = 6.

Thus, the number of zeros at the end of the product is **6**.

### Final Answer
The number of zeros at the end of the product is **6**.