The number of zeros at the end of the product of :
`2^3 xx 3^4 xx 4^5 xx 5^6 + 3^5 xx 5^7 xx 7^9 xx 8^10 + 4^5 xx 5^6 xx 6^7 xx 7^8 - 10^2 xx 15^3 xx 20^4` is :

Options

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

Answer

To find the number of zeros at the end of the product given in the question, we need to determine how many times the number 10 can be factored out from the entire expression. Since \(10 = 2 \times 5\), we need to count the factors of 2 and 5 in the expression and take the minimum of the two counts.

Let's break down the expression step by step:

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

1. **Rewrite \(4^5\)**:
\[ 4^5 = (2^2)^5 = 2^{10} \]

2. **Combine the powers of 2**:
\[ 2^3 \times 2^{10} = 2^{13} \]

3. **Count the powers of 5**:
\[ 5^6 \]

Thus, the first term contributes:
- \(2^{13}\)
- \(5^6\)

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

1. **Rewrite \(8^{10}\)**:
\[ 8^{10} = (2^3)^{10} = 2^{30} \]

2. **Count the powers of 5**:
\[ 5^7 \]

Thus, the second term contributes:
- \(2^{30}\)
- \(5^7\)

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

1. **Rewrite \(4^5\)**:
\[ 4^5 = 2^{10} \]

2. **Rewrite \(6^7\)**:
\[ 6^7 = (2 \times 3)^7 = 2^7 \times 3^7 \]

3. **Combine the powers of 2**:
\[ 2^{10} \times 2^7 = 2^{17} \]

4. **Count the powers of 5**:
\[ 5^6 \]

Thus, the third term contributes:
- \(2^{17}\)
- \(5^6\)

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

1. **Rewrite \(10^2\)**:
\[ 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 \]

2. **Rewrite \(15^3\)**:
\[ 15^3 = (3 \times 5)^3 = 3^3 \times 5^3 \]

3. **Rewrite \(20^4\)**:
\[ 20^4 = (2^2 \times 5)^4 = 2^8 \times 5^4 \]

4. **Combine the contributions**:
- From \(10^2\): \(2^2 \times 5^2\)
- From \(15^3\): \(3^3 \times 5^3\) (ignore \(3^3\))
- From \(20^4\): \(2^8 \times 5^4\)

Thus, the fourth term contributes:
- \(2^{10}\) (from \(2^2 + 2^8\))
- \(5^9\) (from \(5^2 + 5^3 + 5^4\))

### Step 5: Combine all contributions

Now, we combine all the contributions:

1. From the first term: \(2^{13} \times 5^6\)
2. From the second term: \(2^{30} \times 5^7\)
3. From the third term: \(2^{17} \times 5^6\)
4. From the fourth term: \(2^{10} \times 5^9\)

### Total powers of 2 and 5:

- Total power of 2:
\[ 13 + 30 + 17 + 10 = 70 \]

- Total power of 5:
\[ 6 + 7 + 6 + 9 = 28 \]

### Step 6: Determine the number of zeros

The number of zeros at the end of the product is given by the minimum of the total powers of 2 and 5:
\[ \text{Number of zeros} = \min(70, 28) = 28 \]

### Final Answer
The number of zeros at the end of the product is **28**.

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