document.write( "Question 826855: Find the largest three digit number such that the number minus the sum of its digit is a perfect square. \n" ); document.write( "
Algebra.Com's Answer #498331 by Edwin McCravy(20055)\"\" \"About 
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Find the largest three digit number such that the number minus the sum of its digit is a perfect square.
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document.write( "Let h = the hundreds digit\r\n" );
document.write( "Let t = the tens digit\r\n" );
document.write( "Let u = the units (or ones) digit\r\n" );
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document.write( "Then the number = 100h+10t+u\r\n" );
document.write( "Sum of the digits = h+t+u\r\n" );
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document.write( "The number - sum of digits) = perfect square.\r\n" );
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document.write( "100h+10t+u - (h+t+u) = perfect square\r\n" );
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document.write( "100h+10t+u - h-t-u = perfect square\r\n" );
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document.write( "99h+9t = perfect square\r\n" );
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document.write( "This shows that the solution is independent of u.\r\n" );
document.write( "Since we are looking for the largest three digit\r\n" );
document.write( "number, we will therefore take u=9\r\n" );
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document.write( "9(11h+t) = perfect square\r\n" );
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document.write( "So the perfect square must be a square root divisible by 9\r\n" );
document.write( "So its square root must be divisible by 3.\r\n" );
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document.write( "Since V999 = 31.6... the largest 3-digit perfect square is 31² or 961.\r\n" );
document.write( "The largest number that does not exceed 31 that is divisible by 3\r\n" );
document.write( "is 30. so 30 is a candidate for the square root, and 30² or 900 is\r\n" );
document.write( "a candidate for the perfect square we are looking for:\r\n" );
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document.write( "9(11h+t) = 900\r\n" );
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document.write( "   llh+t = 100\r\n" );
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document.write( "       t = 100-11h\r\n" );
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document.write( "All digits are less than 10, so\r\n" );
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document.write( "          t < 10\r\n" );
document.write( "    100-11h < 10\r\n" );
document.write( "       -11h < -90\r\n" );
document.write( "          h > 8.18...\r\n" );
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document.write( "The only digit larger than 8.18... is 9, so h = 9\r\n" );
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document.write( "       t = 100-11(9) = 100-99 = 1\r\n" );
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document.write( "Checking:  919 has sum of digits 9+1+9 = 19 \r\n" );
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document.write( "919-19 = 900 = 30²\r\n" );
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document.write( "Answer: 919\r\n" );
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document.write( "Edwin
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