document.write( "Question 1210504: If A and B are positive 2 digest integers, how many solutions are there of the equation 2a+3b=100 \n" ); document.write( "

Algebra.Com's Answer #853062 by ikleyn(53354) $\"\"$ $\"About$

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\n" ); document.write( "If 'a' and 'b' are positive $\"highlight%28cross%282-digest%29%29\"$ 2-digit integers,
\n" ); document.write( "how many solutions are there of the equation 2a+3b=100.
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\n" ); document.write( "\n" ); document.write( "        The post by @MathLover1 isn't a solution to the problem because it doesn't answer the problem's question.\r
\n" ); document.write( "\n" ); document.write( "        The question \"how many\" remains unanswered, or she shifts the calculation to the reader.\r
\n" ); document.write( "\n" ); document.write( "        Therefore, her post is a talk about a solution, but not the solution itself.\r
\n" ); document.write( "\n" ); document.write( "        I came to deliver the solution as it should be.\r
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document.write( "In  equation\r\n" );
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document.write( "    2a + 3b = 100,\r\n" );
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document.write( "the term '2a' is an even number and right side '100' is an even number, too.\r\n" );
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document.write( "Hence, the term '3b' must be even number.  It implies that number 'b' must be even number.\r\n" );
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document.write( "Then '3b' is a multiple of 6.\r\n" );
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document.write( "Thus, '3b' should be multiple of 6 and 'b' itself should be a two-digit positive integer number.\r\n" );
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document.write( "The smallest such number '3b' is 30.\r\n" );
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document.write( "So, the numbers '3b'  form the set < 30, 36, 42, . . . >.\r\n" );
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document.write( "We want to determine the maximum possible value of '3b'.\r\n" );
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document.write( "It should allow 'a' in equation  '2a = 100 - 3b'  to be a 2-digit positive integer number.\r\n" );
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document.write( "So, the greatest possible value of '3b' is not more than 80.\r\n" );
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document.write( "Thus, our set for '3b' is  the sequense  < 30, 36, 42, . . . , 78 >.\r\n" );
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document.write( "It contains   = 9 numbers.\r\n" );
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document.write( "So, equation  2a + 3b = 100  has 9 solutions in positive integer numbers (a,b),\r\n" );
document.write( "such that 'a' and 'b' are 2-digit positive integer numbers.\r\n" );
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document.write( "ANSWER.  There are 9 solutions under imposed conditions.\r\n" );
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