Square digital sum

Algebra Level 2

The digital sum of the year 2016 is $2+0+1+6 = 9$ , which is a perfect square.

How many years during the twenty-first century have a digital sum that is a perfect square?

The answer is 16.

3 solutions

Pranjal Jain
Dec 11, 2014

First and second digit must be $2$ and $0$ respectively as we are given that it is $21^{st}$ century. Let $a$ and $b$ denote other two digits.

Sum $=4$ , $a+b=2\Rightarrow$ 3 possible ways $(0,2),(1,1),(2,0)$
Sum $=9$ , $a+b=7\Rightarrow$ 8 possible ways $(0,7)(1,6)...(7,0)$
Sum $=16$ , $a+b=14\Rightarrow$ 5 possible ways $(5,9)(6,8)...(9,5)$ $\color{#D61F06}{\text{Since } 0 \leq\text{a,b} \leq 9}$

Total possible ways= $3+8+5=\boxed{16}$

and the solution is a square too! nice.

Aloysius Ng - 6 years, 5 months ago

I didn't noticed that!! Cool!

Pranjal Jain - 6 years, 5 months ago

Do not forget that $2100$ is part of the $21^{\text{st}}$ century was well, so the second digit isn't necessarily $0,$ but in the case that it is $1,$ there are no solutions.

Caleb Townsend - 6 years, 2 months ago

Ahsanul Habib
Dec 14, 2014

2001-2099

Highest sum= 2+0+9+9=20
Squares could be=4, 9 and 16
20---02, 11, 20 =total 3
20---07, 16, 25, 34, 43, 52, 61, 70 =total 8
20---59, 68, 77, 86, 95 =total 5
So in total= 16

Tushar Kaushik
Dec 13, 2014

2002, 2011, 2020
2007, 2016, 2025, 2034, 2043, 2052, 2061, 2070
2059, 2068, 2077, 2086, 2095
only these have their sum as perfect square

Square digital sum

The answer is 16.

3 solutions

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