Number Theory


Factorials Warmup


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

What is the smallest positive integer nn such that n!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 nn such that n!n! is divisible by 9?9?

Note: If nn is a positive integer, n!=1×2×3××(n1)×n.n! = 1 \times 2 \times 3 \times \cdots \times (n - 1) \times n. For example, 5!=1×2×3×4×5.5! = 1 \times 2 \times 3 \times 4 \times 5.

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

x!=3!×5!x! = 3! \times 5! What is x?x?


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