Doudou_Chen wrote:

@above isn't $1296=2^4\cdot3^4$ have $(4+1)(4+1)=25$ divisors, so it also works, but is smaller? I don't think 2310 is correct. (I'm not sure 1296 is correct either, but it's a counterexample.)

EDIT: $720=2^4\cdot3^2\cdot5^1$ has $(4+1)(2+1)(1+1)=30$ , so 1296 is not correct. There may be even smaller ones though...

To make a number smaller can make $3^2$ into $2^2$ making number into $2^6*5$ but that'll make number of divisors $(6+1)(1+1)=14$ or make just one 3 into 2 making $(5+1)(1+1)(1+1)=24$ divisprs the other way to reduce number is to make $3^2$ into one 5 because 3*3 = 9>5. but that'll make $(4+1)(2+1)=15$ divisors. We could also make 5 into 2 or 3 $(5+1)(2+1)=18 or (4+1)(3+1)=20$ divisors. So it sure seems like 720 is the smallest such number