Find the number of zeros at the end in product `5^6 .6^7 .7^8 .8^9 .9^(10) .30^(31)` .

Explanation

To find the number of zeros at the end of the product \(5^6 \cdot 6^7 \cdot 7^8 \cdot 8^9 \cdot 9^{10} \cdot 30^{31}\), we need to determine how many times the product can be divided by 10. Since \(10 = 2 \cdot 5\), we need to find the minimum of the number of factors of 2 and the number of factors of 5 in the product.

### Step-by-Step Solution:

1. **Break down each term into prime factors**:
- \(5^6\) remains \(5^6\).
- \(6^7 = (2 \cdot 3)^7 = 2^7 \cdot 3^7\).
- \(7^8\) remains \(7^8\).
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