October 14, 2002

Factorial Factoring
**Dr. Weevil** at October 14, 2002 05:11 PM

Sasha Volokh has some interesting mathematical riddles on the Volokh Experience. I think I can answer the first one: why does 29! (29 factorial, or 1 x 2 x 3 x 4 . . . x 29) end with 6 zeroes? Because every time you multiply a number by 2 and by 5 you add another zero. (I don't have to prove that, do I?) There are 14 even numbers in 29!, some of which are multiples of higher powers of 2, so there is no shortage of 2s among the smallest factors of the humongous (31-digit) total. But there are exactly six 5s in 29!: one each in 5, 10, 15, and 20, and two in 25. So what's my prize? (I'll think about the related puzzles when I'm feeling less sluggish.)

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