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?
Answers
The total number of years having a square digit sum during the twenty first century = 16.
The 21st century starts from 2001 - 2100.
So, starting with 2001
2001 = 2 + 0 + 0 + 1 = 3
2002 = 2 + 0 + 0 + 2 = 4 (square number)
2003 = 2 + 0 + 0 + 3 = 5
2004 = 2 + 0 + 0 + 4 = 6
2005 = 2 + 0 + 0 + 5 = 7
2006 = 2 + 0 + 0 + 6 = 8
2007 = 2 + 0 + 0 + 7 = 9 (square number)
2008 = 2 + 0 + 0 + 8 = 10
2009 = 2 + 0 + 0 + 9 = 11
2010 = 2 + 0 + 1 + 0 = 3
2011 = 2 + 0 + 1 + 1 = 4 (square number)
2012 = 2 + 0 + 1 + 2 = 5
2013 = 2 + 0 + 1 + 3 = 6
2014 = 2 + 0 + 1 + 4 = 7
2015 = 2 + 0 + 1 + 5 = 8
2016 = 2 + 0 + 1 + 6 = 9 (square number)
2017 = 2 + 0 + 1 + 7 = 10
2018 = 2 + 0 + 1 + 8 = 11
2019 = 2 + 0 + 1 + 9 = 12
2020 = 2 + 0 + 2 + 0 = 4 (square number)
2021 = 2 + 0 + 2 + 1 = 5
2022 = 2 + 0 + 2 + 2 = 6
2023 = 2 + 0 + 2 + 3 = 7
2024 = 2 + 0 + 2 + 4 = 8
2025 = 2 + 0 + 2 + 5 = 9 (square number)
2026 = 2 + 0 + 2 + 6 = 10
2027 = 2 + 0 + 2 + 7 = 11
2028 = 2 + 0 + 2 + 8 = 12
2029 = 2 + 0 + 2 + 9 = 13
2030 = 2 + 0 + 3 + 0 = 5
2031 = 2 + 0 + 3 + 1 = 6
2032 = 2 + 0 + 3 + 2 = 7
2033 = 2 + 0 + 3 + 3 = 8
2034 = 2 + 0 + 3 + 4 = 9 (square number)
2035 = 2 + 0 + 3 + 5 = 10
2036 = 2 + 0 + 3 + 6 = 11
2037 = 2 + 0 + 3 + 7 = 12
2038 = 2 + 0 + 3 + 8 = 13
2039 = 2 + 0 + 3 + 9 = 14
2040 = 2 + 0 + 4 + 0 = 6
2041 = 2 + 0 + 4 + 1 = 7
2042 = 2 + 0 + 4 + 2 = 8
2043 = 2 + 0 + 4 + 3 = 9 (square number)
2044 = 2 + 0 + 4 + 4 = 10
2045 = 2 + 0 + 4 + 5 = 11
2046 = 2 + 0 + 4 + 6 = 12
2047 = 2 + 0 + 4 + 7 = 13
2048 = 2 + 0 + 4 + 8 = 14
2049 = 2 + 0 + 4 + 9 = 15
2050 = 2 + 0 + 5 + 0 = 7
2051 = 2 + 0 + 5 + 1 = 8
2052 = 2 + 0 + 5 + 2 = 9 (square number)
2053 = 2 + 0 + 5 + 3 = 10
2054 = 2 + 0 + 5 + 4 = 11
2055 = 2 + 0 + 5 + 5 = 12
2056 = 2 + 0 + 5 + 6 = 13
2057 = 2 + 0 + 5 + 7 = 14
2058 = 2 + 0 + 5 + 8 = 15
2059 = 2 + 0 + 5 + 9 = 16 (square number)
2060 = 2 + 0 + 6 + 0 = 8
2061 = 2 + 0 + 6 + 1 = 9 (square number)
2062 = 2 + 0 + 6 + 2 = 10
2063 = 2 + 0 + 6 + 3 = 11
2064 = 2 + 0 + 6 + 4 = 12
2065 = 2 + 0 + 6 + 5 = 13
2066 = 2 + 0 + 6 + 6 = 14
2067 = 2 + 0 + 6 + 7 = 15
2068 = 2 + 0 + 6 + 8 = 16 (square number)
2069 = 2 + 0 + 6 + 9 = 17
2070 = 2 + 0 + 7 + 0 = 9 (square number)
2071 = 2 + 0 + 7 + 1 = 10
2072 = 2 + 0 + 7 + 2 = 11
2073 = 2 + 0 + 7 + 3 = 12
2074 = 2 + 0 + 7 + 4 = 13
2075 = 2 + 0 + 7 + 5 = 14
2076 = 2 + 0 + 7 + 6 = 15
2077 = 2 + 0 + 7 + 7 = 16 (square number)
2078 = 2 + 0 + 7 + 8 = 17
2079 = 2 + 0 + 7 + 9 = 18
2080 = 2 + 0 + 8 + 0 = 10
2081 = 2 + 0 + 8 + 1 = 11
2082 = 2 + 0 + 8 + 2 = 12
2083 = 2 + 0 + 8 + 3 = 13
2084 = 2 + 0 + 8 + 4 = 14
2085 = 2 + 0 + 8 + 5 = 15
2086 = 2 + 0 + 8 + 6 = 16 (square number)
2087 = 2 + 0 + 8 + 7 = 17
2088 = 2 + 0 + 8 + 8 = 18
2089 = 2 + 0 + 8 + 9 = 19
2090 = 2 + 0 + 9 + 0 = 11
2091 = 2 + 0 + 9 + 1 = 12
2092 = 2 + 0 + 9 + 2 = 13
2093 = 2 + 0 + 9 + 3 = 14
2094 = 2 + 0 + 9 + 4 = 15
2095 = 2 + 0 + 9 + 5 = 16 (square number)
2096 = 2 + 0 + 9 + 6 = 17
2097 = 2 + 0 + 9 + 7 = 18
2098 = 2 + 0 + 9 + 8 = 19
2099 = 2 + 0 + 9 + 9 = 20
2100 = 2 + 1 + 0 + 0 = 3
∴ The total number of years having a square digit sum during the twenty first century = 16.
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