document.write( "Question 580761: How many of the numbers from 10 through 85 have the sum of their digits equal to a perfect square? \n" ); document.write( "

Algebra.Com's Answer #371465 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The sum of the digits would be at least 1 (as for 10, which I am assuming is included). It could be at most 18 for a 2 digit number (as in 99), but since the highest we can go is 85, we may not get to 18 with the sum of the digits.
\n" ); document.write( "The only perfect squares between 1 and 18 are 1, 4, 9, and 16.
\n" ); document.write( "The only number between 10 and 85 whose digits add up to 1 is $\"highlight%2810%29\"$ .
\n" ); document.write( "The only numbers between 10 and 85 whose digits add up to 4 are:
\n" ); document.write( " $\"highlight%2813%29\"$ , $\"highlight%2822%29\"$ , $\"highlight%2831%29\"$ , and $\"highlight%2840%29\"$ (4 numbers).
\n" ); document.write( "The only numbers between 10 and 85 whose digits add up to 9 are:
\n" ); document.write( " $\"highlight%2818%29\"$ , $\"highlight%2827%29\"$ , $\"highlight%2836%29\"$ , ... and $\"highlight%2881%29\"$ (8 numbers).
\n" ); document.write( "The only number between 10 and 85 whose digits add up to 16 is $\"highlight%2879%29\"$ (88 and 97 are too high).
\n" ); document.write( "So I am counting $\"1%2B4%2B8%2B1=14\"$ numbers whose digits add up to a perfect square.
\n" ); document.write( "My answer is $\"highlight%2814%29\"$ . \n" ); document.write( "

\n" );