document.write( "Question 1179510: A certain two-digit number is equivalent to five times the sum of the digits.It is found to be 9 less than the number formed when the digits are interchanged.Find the number. \n" ); document.write( "

Algebra.Com's Answer #809086 by ikleyn(53937) $\"\"$ $\"About$

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\n" ); document.write( "A certain two-digit number is equivalent to five times the sum of the digits.
\n" ); document.write( "It is found to be 9 less than the number formed when the digits are interchanged.Find the number.
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document.write( "Let \"a\" be the tens digit and \"b\" be the ones digit.\r\n" );
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document.write( "Then the number is 10a+b,  and the first condition gives this equation\r\n" );
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document.write( "    10a + b = 5(a+b),      (1)\r\n" );
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document.write( "which implies\r\n" );
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document.write( "    10a + b = 5a + 5b\r\n" );
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document.write( "     5a     = 4b.          (2)\r\n" );
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document.write( "The solution a = b = 0 is excluded (since \"a\" can not be zero, as a leading digit).\r\n" );
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document.write( "Then for \"a\" and \"b\" being digits in the interval from 1 to 9, equation (1) has one \r\n" );
document.write( "and only one solution in integer numbers  a = 4, b = 5.\r\n" );
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document.write( "So, the number is 45, and it is a UNIQUE number, satisfying the first imposed condition.\r\n" );
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document.write( "Thus the second condition is EXCESSIVE and is non-necessary.\r\n" );
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document.write( "It adds nothing to the problem, but we still need check that it is satisfied and \r\n" );
document.write( "does not contradict to the first condition.\r\n" );
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document.write( "Fortunately, it is so, and the ANSWER is  45.\r\n" );
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