The smallest number with 2013 divisors is:2 ^{60} x 3^{10} x 5^{2} = 1701971548138242677145600Congratulations 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 = 2
^{2}x 3 - 363 = 3 x 11 x 11 = 3 x 11
^{2} - 5940 = 2 x 2 x 3 x 3 x 3 x 5 x 11 = 2
^{2}x 3^{3}x 5^{1}x 11^{1}
The divisors can be found by combining all the possible products of the prime factors. To show this, let's consider 120 = 2 ^{3} x 3^{1} x 5^{1 }, 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:
(To understand this table, you need to know that n ^{0} = 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 = p ^{a} x q^{b} x r^{c} 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 5 ^{2} x 3^{10} x 2^{60} = 1701971548138242677145600Is 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. 2 ^{60} x 3^{32} = 21363868241979282569605526228828162 ^{182} x 3^{10} = 3619693167703594847888081344247944279340117057032680995624962 ^{671} x 3^{2} = ...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! |