document.write( "Question 1138622: Find all positive integers a and b so that (a+1)/b and (b+2)/a are simultaneously
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Algebra.Com's Answer #756624 by Edwin McCravy(20060)\"\" \"About 
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document.write( "greenestamps was looking for:\r\n" );
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\n" ); document.write( "Find all positive integers a and b so that (a+1)/b and (b+2)/a are simultaneously
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document.write( "\"matrix%281%2C4%2C%28a%2B1%29%2Fb%2C%22%22%2C%22%22%2C+%28b%2B2%29%2Fa%29%29\"\r\n" );
document.write( "Those are positive integers, so\r\n" );
document.write( "\"system%281%3C=%28a%2B1%29%2Fb%2C+1%3C=%28b%2B2%29%2Fa%29\"\r\n" );
document.write( "\"system%28b%3C=a%2B1%2C+a%3C=b%2B2%29\"\r\n" );
document.write( "Add 1 to both sides of the second inequality\r\n" );
document.write( "\"system%28b%3C=a%2B1%2C+a%2B1%3C=b%2B3%29\"\r\n" );
document.write( "Those inequalities can be combined as\r\n" );
document.write( "\"b%3C=a%2B1%3C=b%2B3\"\r\n" );
document.write( "So we have four cases: a+1=b, a+1=b+1, a+1=b+2, a+1=b+3\r\n" );
document.write( "which simplify to a=b-1, a=b, a=b+1, a=b+2\r\n" );
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document.write( "Case 1:  a=b-1 substitute in \r\n" );
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document.write( "For the expression to be a positive integer,\r\n" );
document.write( "b-1 is either 1 or 3.\r\n" );
document.write( "For b-1=1 or b=2 and a=b-1=2-1=1, (a,b)=(1,2)\r\n" );
document.write( "For b-1=3 or b=4 and a=b-1=4-1=3, (a,b)=(3,4)\r\n" );
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document.write( "Case 2:  a=b substitute in \r\n" );
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document.write( "Those will both be integers only if b=1. \r\n" );
document.write( "a=b=1, (a,b)=(1,1) \r\n" );
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document.write( "Case 3:  a=b+1 substitute in \r\n" );
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document.write( "For the second expression to be a positive integer,\r\n" );
document.write( "b+1 must be 1, which makes b=0. So we must discard \r\n" );
document.write( "Case 3 as impossible. \r\n" );
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document.write( "Case 4:  a=b+2 substitute in \r\n" );
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document.write( "For the first expression to be a positive integer,\r\n" );
document.write( "b is either 1 or 3.\r\n" );
document.write( "For b=1, a=b+2=1+2=3, so (a,b)=(3,1) is a solution.\r\n" );
document.write( "For b=3, a=b+2=3+2=5, so (a,b)=(5,3) is a solution.\r\n" );
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document.write( "So there are 5 solutions:\r\n" );
document.write( "(a,b)=(1,2)\r\n" );
document.write( "(a,b)=(3,4) \r\n" );
document.write( "(a,b)=(1,1)\r\n" );
document.write( "(a,b)=(3,1)\r\n" );
document.write( "(a,b)=(5,3)\r\n" );
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document.write( "Edwin
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