Prime factors in n! (e.g. how many 2s in 45!)

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Learn it To count how many times a prime p divides n!, add ⌊n ÷ p⌋ + ⌊n ÷ p²⌋ + ⌊n ÷ p³⌋ + … (Legendre's formula). Trailing zeros of n! = the number of 5s.

Example: How many 2s are in 45!?

Multiples of 2: ⌊45 ÷ 2⌋ = 22. Of 4: 11. Of 8: 5. Of 16: 2. Of 32: 1.
Each extra power gives one more 2.
Total: $22 + 11 + 5 + 2 + 1 = 41$.

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