Q. The digit sum of the year 2007 is 2 + 0 + 0 + 7 = 9, which is a square number. How many years during the twenty-first century have a square digital sum?

• The twenty-first century runs from the year 2001 to 2100 inclusive, so with the
  exception of the year 2100, for which the digital sum is three anyway, the first two
  digits of all the other years are fixed as 20. Hence for the sum of all the digits to be
  square the sum of the last two digits number be two less than a perfect square.
  As the maximum sum of the last two digits is 18, the greatest sum will be 20.
  Therefore we can obtain digital sums of 4, 9, or 16, with the last two digits adding to
  2, 7, or 14 respectively.
  4: 2002, 2011, 2020
  9: 2007, 2016, 2025, 2034, 2043, 2052, 2061, 2070
  16: 2059, 2068, 2077, 2086, 2095
  That is, there are sixteen years during the twenty-first century for which the digital
  sum is square.
  
• 11 years agoHelpfull: Yes(125) No(1)
• very nice shelly. thanks for ur ans.
  
• 11 years agoHelpfull: Yes(2) No(0)
• in 21 century 20-- is fixed.now we have to find sum last two digits. the sum should be 2,7,14 to be perfect square. so for sum 2-3 ways,sum 7-8 ways, 14-5 ways
  total =16 years
• 10 years agoHelpfull: Yes(2) No(0)