If $2004$ is split after the third digit into a three-digit integer and a one-digit integer, then the two integers, $200$ and $4
$, have a common factor greater than one. The years $2005$ and $2006$ each have this same property, too. What is the first odd-numbered year after $2006$ that has this property?
We note that 2013 has the property: since the sum of the digits of 201, 2+0+1=3, is divisible by 3, 201 is itself divisible by 3. We must now check whether any odd-numbered years before 2013 have the property. 200 is not divisible by the prime 7, so 2007 does not have this property. Likewise, the sum of the digits of 200 is 2, which is not divisible by 3, so 200 and 9=3^2 are relatively prime. Clearly 2011 does not have the desired property since every natural number is relatively prime to 1. Thus, the first odd-numbered year after 2006 that has the desired property is 2013.
