First you need to look at the prime factorization of a number meaning you need to break down into prime numbers.
Then you look at the exponents of the number, 360= 2^3 *3^2*5.
Increase the powers by 1,(3+1)=4,(2+1)=3 and (1+1)=2.
Then you multiply all of them.
The number of factors of 360= (3+1)(2+1)(1+1)=24.
A number which has 14 factors, 14 = 7*2, a number of the form x^6* y or a number of the form z^13,where x,y,z are prime,the number is 2^6*3 has 14 factors as long as x does not equal y.
A number which has 15 factors must be of the form x^2*y^4 or z^14,where (x,y,z) are prime,the number is 3^2*2^4.
18 factors, we can think about the prime factorization of 18, 18 = 3^2*2, 18 = 3*3*2, a^2*b^2*c,where (a,b,c) are prime,a=2,b=3,c=5, 2^2*3^2*5.
Only square numbers have an odd amount of factors.The prime factorization of a square number has to have only even numbers in the exponents and if you add 1 to an even it becomes an odd, odd*odd*odd*odd....=odd.
Smallest number with 100 factors,100=2*2*5*5, 100=2^2*5^2,5-1=4,2-1=2, 2^4*3^4*5*7 is the smallest number with 100 divisors.
The number with the most factors which is below 1000,by trial and improvement you get 840 which has 32 factors.If we go over 32 then the number we chose has to be bigger than 1000.