60? :D
Why the question mark, you don't know for sure? Is that's what's tripping people up? is there a way to be sure without a brute-force method by excluding all possible candites. Obviosly there would be. In this case it's also worth thinking about. Slightly more difficult challenge.
Anyway, the point is: Ultimately that's the practical reason why we use 360° degrees as well, because it's a small-ish number divisible by all digits save the 7.
Correction 360 isn't divisible by 7 or 8. But still it's a small and by far most divisible number. But yeah, to the initial question 60 is the right answer, which is why we use and continue using 60 min and 60 seconds, and perhaps even a 12 hour clockface (divisible by 2,3,4).
To determine the smallest number divisible by any series of numbers we just need to write them down 10|9|8|7|6|5|4|3|2|1, and then use only the biggest multipliers and strike all divisors/prime-factors, so for a 10 we can strike 5 and 2, 9 strikes 3, 8 strikes 2 and 4, 6 strikes 2 and 3 .... leaving only 10*9*8*7*6 = 30'240.
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Why is it so difficult for people to figure out? Why is this difficult for a child (without giving tips), who already knows division. Is he or she just lazy? I though it would be a neat way to make someone discover something true about convention and life on their own. But maybe I need to rephrase the question.