special numbers, divisibility tests and prime factorization

Here is the first blog entry for this site on the first day of 2013. My new year's resolution is to try and compose at least two blog entries for each week of the year. A bit of a challenge but we'll see how it goes.

Anything interesting or special about the number 2013? Of course, it's not a prime number since the sum of its digits is 6, a multiple of 3, thereby indicating it must be divisible by 3.

I use a couple of websites and one particularly 'interesting' book when looking for any special characteristics of a number.  The book is The Penguin Dictionary of Curious and Interesting Numbers by David Wells, and the websites are Zoo of Numbers (Archimedes Laboratory) and What's Special About This Number? The main feature of all three of these sources is that they give an ascending list of numbers with some 'special' or 'interesting' characteriscs for each.  2013 does not appear in any of these sources, so I'm not aware of anything 'interesting' about the number 2013, but perhaps there might be for its prime factors.

2013 = 3 x 671

And 671 is divisible by 11.  How do we know that?  The rule for testing divisibility by 11 is not as well known as the rule for divisibility by 3, but it's nearly as easy to apply. First, find the sum of the odd numbered digits and then subtract from this sum the sum of the even numbered digits.  If the result is either 0 or a number divisible by 11, then the original number is divisible by 11.  Consider 671: since (6+1) - 7 = 0 then 671 is divisible by 11.

2013 = 3 x 11 x 61

Since 3, 11 and 61 are all prime numbers then 3 x 11 x 61 is the unique prime factorization of 2013.  This makes me think of two (unrelated) questions: (1) can we find something 'interesting' about the numbers 3, 11 and 61 ?, and (2) how do we know that the prime factorization of a number is unique ?  Teachers routinely tell students that a number's prime factorization is unique. It seems fairly obvious or intuitive, but is it?

(1)  The sources I mentioned give several interesting features for the numbers 3, 11 and 61. Here is one unique feature of each:
      3 is the only prime number that is one less than a perfect square
      11 is the only palindromic (same forwards as backwards) prime number with an even number of digits
      61 is the only prime number whose reversal (16) is a perfect square

(2)  It is true that the prime factorization of a number is unique. I've always told my students this fact without any hesitation and thought it fairly obvious - until I read one of the chapters in the fascinating book The Enjoyment of Math written by Hans Rademacher and Otto Toeplitz and published by Princeton University Press in 1957. It's a little gem of a book. Take a look at chapter 11 entitled Is the Factorization of a Number into Prime Factors Unique?

I hope that the year 2013 is special and interesting for you