For a positive integer n, define S(n) to be the sum of the positive divisors of n. Which of the
following is the smallest?
Choose the correct choice from the following options: A) S(2010) B) S(2011) C) S(2012) D) S(2013) E) S(2014)
2011 is prime, so that S(2011) = 1 + 2011= 
<gadget id="calculator">1 + 2011</gadget>
<output>2_012</output>
2012, the smallest.   But this answer can be verified without checking that 2011 is prime. Observe that   S(2010) &gt; 2010 + 1005 &gt; 3000   S(2013) &gt; 2013 + 671 &gt; 2600   and S(2012) and S(2014) are also larger than 3000.   Check by hand that 2, 3, 5, 7, and 11 do not divide 2011, and so if 2011 is not prime, then it   is a product of two primes p and q. If p 6= q, then   S(2011) = 2011 +   2011/p   +   2011/q   + 1;   and p and q are greater than 11, so that this is smaller than 2600. If p = q then this is smaller   still.   correct answer S(2011)
<result>B</result>
For a positive integer n, define S(n) to be the sum of the positive divisors of n. Which of the
following is the smallest?