The smallest number with 2013 divisors is:

260 x 310 x 52 = 1701971548138242677145600

Congratulations if you worked this out for yourself!

Step-by-step Solution:
Each integer has a unique prime factorization. For example,
• 12 = 2 x 2 x 3 = 22 x 3
• 363 = 3 x 11 x 11 = 3 x 112
• 5940 = 2 x 2 x 3 x 3 x 3 x 5 x 11 = 22 x 33 x 51 x 111
The divisors can be found by combining all the possible products of the prime factors. To show this, let's consider 120 = 23 x 31 x 51 , which has 16 divisors: 1,2,3,4,5,6,10,12,15,20,24,30,40,60,120. We can write out these divisors as follows:

 1 = 20 x 30 x 50 2 = 21 x 30 x 50 3 = 20 x 31 x 50 4 = 22 x 30 x 50 5 = 20 x 30 x 51 6 = 21 x 31 x 50 8 = 23 x 30 x 50 10 = 21 x 30 x 51 12 = 22 x 31 x 50 15 = 20 x 31 x 51 20 = 22 x 30 x 51 24 = 23 x 31 x 50 30 = 21 x 31 x 51 40 = 23 x 30 x 51 60 = 22 x 31 x 51 120 = 23 x 31 x 51

(To understand this table, you need to know that n0 = 1 for any value of n).

Do you see how all possible combinations appear? There are four possibilities for the power of 2 (i.e. 0,1,2, and 3), and two possibilities each for the power of 3 and 5 (i.e. 0 and 1). So there are 16 divisors in total, as 4 x 2 x 2 = 16.

Now let's consider a general number n with prime factorization

n = pa x qb x rc x ...

where p, q, r, etc. are prime numbers, and a, b, c etc. are integers (e.g. 0,1,2,3, ...). How many divisors does this number have? Following the argument above, it must have

(a + 1) x (b + 1) x (c + 1) ...

divisors in total. This formula is the key to answering the problem.

Now, we need to find the prime factorization of 2013:

2013 = 3 x 11 x 61.

To find a number with 2013 divisors, we can set (a + 1) = 3, (b + 1) = 11, (c + 1) = 61, so that a = 2, b = 10, c = 60

These are the indices. To find the smallest number with 2013 divisors, we should build our number from the smallest possible primes factors, 5, 3, and 2, so

52 x 310 x 260 = 1701971548138242677145600

Is this really the smallest possible number with 2013 divisors? Well, we could try using other (non-prime) factorizations of 2013, e.g., 33 x 61, 11 x 183, 3 x 671.  This leads to other solutions, e.g.

260 x 332 = 2136386824197928256960552622882816
2182 x 310 = 361969316770359484788808134424794427934011705703268099562496
2671 x 32 = ...

But these are all clearly bigger than the first solution we found.

One aim of this Mini-Challenge was to show that there are some very "simple" mathematical questions which cannot be solved easily with a naive computer code. We still need to use our brains!