The product of the first seven positive integers is equal to \[7 \times 6\times 5\times 4\times 3\times 2\times 1= 5040\] Mathematicians will write this product as \(7!\). This is read as “\(7\) factorial”. So, \(7! = 7 \times 6\times 5\times 4\times 3\times 2\times 1= 5040\).
This factorial notation can be used with any positive integer. For example, \(11! = 11 \times 10 \times 9 \times \cdots \times 3 \times 2 \times 1 = 39\,916\,800\). The three dots “\(\cdots\)” represent the product of the integers between \(9\) and \(3\).
In general, for a positive integer \(n\), \(n!\) is equal to the product of the positive integers from \(1\) to \(n\).
Find the smallest positive integer \(n\) such that \(n!\) ends in exactly six zeros.
Theme: Number Sense