document.write( "Question 1172751: The five-digit numbers 8a2bc and 3a8bc are both perfect squares. What is the product of a x c?\r
\n" ); document.write( "\n" ); document.write( "Previous attempts: Using square rules (If c = 5, then b MUST = 2). Narrowing down possible endings to square endings: 1, 4, 5, 6, 9
\n" ); document.write( "Difficulty: Unable to find letter a, as well as find substitutions that work for both five-digit numbers\r
\n" ); document.write( "\n" ); document.write( "Thanks in advance! \n" ); document.write( "

Algebra.Com's Answer #797869 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "The three missing digits are the same in both numbers.

\n" ); document.write( "So, given that the two numbers are m^2 and n^2, look at m^2-n^2 = (m+n)(m-n):

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document.write( "     m^2 = 8a2bc\r\n" );
document.write( "     n^2 = 3a8bc\r\n" );
document.write( " ----------------\r\n" );
document.write( " m^2-n^2 = 49400\r\n" );
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document.write( "(m+n)(m-n) = 49400\r\n" );
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document.write( "Look for a factorization of 49400 into the product of two integers of the form m+n and m-n:\r\n" );
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document.write( "(m+n)(m-n) = 49400 = (494)(100)\r\n" );
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document.write( "m+n = 494; m-n = 100 --> m = 297, n = 197\r\n" );
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document.write( "297^2 = 88209\r\n" );
document.write( "197^2 = 38809\r\n" );
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document.write( "a = 8; b = 0; c = 9

\n" ); document.write( "ANSWER: ac = 8*9 = 72

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\n" ); document.write( "NOTE....

\n" ); document.write( "m^2=8a2bc is greater than 80000; that means m is between sqrt(80000) = 282 and 300;

\n" ); document.write( "n^2=3a8bc is between 30000 and 40000; that means n is between sqrt(30000) = 173 and 200

\n" ); document.write( "In the above solution, there are many other factorizations of 49400 into the product of two integers of the form m+n and m-n; but none of the others produces values of m and n that meet those requirements.

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