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Digital Square Sum

Problem

The digital 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?


Solution

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.

Which century since 1 A.D. has the most square digital sums?

Problem ID: 320 (14 Apr 2007)     Difficulty: 1 Star

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