154368

Because n to the 0th, 1st, 2nd... mth power is divisible by n to the mth power, if the prime factorization of a number is (Prime 1)^a*(Prime 2)^b*(Prime 3)^c...(Prime ?)^x, where all of Prime 1, 2, 3, 4... ? are unique, there are (a+1)*(b+1)*(c+1)*(d+1)...*(x+1).
Thus, since 360's prime factorization is 2^3*3^2*5(^1), the number of factors in 360 are (3+1)*(2+1)*(1+1)=4*3*2=24 factors. We can do the same thing with any number. For example, 312500=5^7*2^2, the number of factors are (7+1)*(2+1)=8*3=24. Let's find a number with 14 factors. Let's first find the distinct sets of numbers that multiply to 14, where every number is greater than 1 (because if at least 1 number is one, then there will be a power of 1-1=0th power, which is always 1, and every number times one is itself): (2, 7) and (14). This means The numbers are in the form a^1*b^6, or a^13, where a and b are unique primes. The smallest from of a^1*b^6 is 3^1*2^6=3*64=192 (the way you can do this is by putting the smallest prime on the smallest exponent, then putting the next smallest prime on the next smallest exponent and so on.) and the smallest form of a^13=2^13=8192. 192<8192. So, the smallest number that has 14 factors is 192. Let's repeat this for 15. Sets of numbers: (3,5), (15). smallest for a^2*b^4: 3^2*2^4=144. 2^14=16384. Obviously, 144<16384. The smallest number with 15 factors is 144. We notice 144 is a square number. We'll get back to this later. If we do the same thing with 18 (every sets possible: (2,9), (3,6), (2,3,3), (18)), 188 is the smallest number with 188 numbers. Okay. Now there are an odd number of factors if and only if the number is a square number. Because a product of some integers with at lest even number is even, all of a+1, b+1, c+1... x+1 must be odd, meaning a, b, c... x must be even. And an integer to a even positive power is always a square. This is why only square numbers have an odd number of factors.