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Number Theory

Factorials

Factorials Warmup

         

Which of the following is equal to \(\Large \frac{6!}{6}\)?

What is the smallest positive integer \(n\) such that \(n!\) has exactly 1 trailing zero?

Note: Trailing zeros are sequences of zeros that come at the end of a number. For example, 1,000 has 3 trailing zeros and 1,001 has no trailing zeros.

What is the smallest value of \(n\) such that \(n!\) is divisible by 9?


Note: If \(n\) is a positive integer, \[n! = 1 \times 2 \times 3 \times \cdots \times (n - 1) \times n.\] For example, \[5! = 1 \times 2 \times 3 \times 4 \times 5.\]

Are there 3 consecutive positive integers whose product is not divisible by \(3! \, ?\)

\[x! = 3! \times 5!\] What is \(x?\)

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