Factorials, divisibility and digit puzzles

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Learn it Factorials contain every number up to n as a factor. From 5! on, every factorial ends in 0, which helps with ones-digit questions.

Example: What is the smallest n so that n! is divisible by 48?

$48 = 2^4 \times 3$.
$4! = 24 = 2^3 \times 3$ (only three 2s). $5! = 120 = 2^3 \times 3 \times 5$ (still three).
$6! = 720 = 2^4 \times 3^2 \times 5$ ✓. So n = 6.

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