document.write( "Question 1027853: Five different positive integers added two at a time give the
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Algebra.Com's Answer #643110 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Five different positive integers added two at a time give the following sums: 16, 20, 22, 23, 25, 28, 29, 30, 34, and 37.
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\n" ); document.write( "\n" ); document.write( "I agree with Mr. Edwin McCravy that my solution was wrong.
\n" ); document.write( "Mr. Edwin McCravy, thank you for your critics and detecting the error.\r
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\n" ); document.write( "\n" ); document.write( "I still keep my old solution below for documentary purposes (but probably will delete it later).\r
\n" ); document.write( "\n" ); document.write( "Instead, I will put here the other solution, which is shorter.\r
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document.write( "Let the numbers be a < b < c < d < e\r\n" );
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document.write( "Then the smallest two, a and b, must have sum 16,\r\n" );
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document.write( "a + b = 16\r\n" );
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document.write( "and the largest two, d and e, must have sum 37,\r\n" );
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document.write( "d + e = 37.\r\n" );
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document.write( "   ----------------------------------------------------------------------\r\n" );
document.write( "   Previous 5 lines are exactly from Mr. McCravy solution, but what follows is different.\r\n" );
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document.write( "Then the middle number c is equal to 66 - (a+b) - (d+e) = 66 - 16 - 37 = 13.\r\n" );
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document.write( "(The number 66 goes from my old solution which is correct to this point).\r\n" );
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document.write( "OK. So, the two smallest numbers a and b give the sum of 16, and the third number is 13.\r\n" );
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document.write( "It means that the next sum, 20 (see the condition) is  a+c.\r\n" );
document.write( "It can not be nothing else.\r\n" );
document.write( "If  a + c = 20  and  c = 13, then  a = 20 - 13 = 7. \r\n" );
document.write( "It implies b = 16-7 = 9.\r\n" );
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document.write( "So, the first three numbers are  a = 7, b = 9  and  c = 13.\r\n" );
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document.write( "OK. Now we can make the similar analysis from the other end.\r\n" );
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document.write( "So, the two largest numbers d and e give the sum of 37, and the third number in the descending order is 13.\r\n" );
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document.write( "It means that the next from the largest sum, 34 (see the condition) is  e+c.\r\n" );
document.write( "It can not be nothing else.\r\n" );
document.write( "If  e + c = 34  and  c = 13, then  e = 34 - 13 = 21. \r\n" );
document.write( "It implies d = 37-21 = 16.\r\n" );
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document.write( "Thus the numbers are 7, 9, 13, 16 and 21.\r\n" );
document.write( "Their product is 275184.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Below is my old >>>wrong<<< solution.
\n" ); document.write( "It is still valid in the part that the sum of the five numbers is 66.
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document.write( "Let  ,  ,  ,    and    be our five different positive integers.\r\n" );
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document.write( "We are given that\r\n" );
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document.write( " +  = 16,     (1)\r\n" );
document.write( " +  = 20,     (2)\r\n" );
document.write( " +  = 22,     (3)\r\n" );
document.write( " +  = 23,     (4)\r\n" );
document.write( " +  = 25,     (5)\r\n" );
document.write( " +  = 28,     (6)\r\n" );
document.write( " +  = 29,     (7)\r\n" );
document.write( " +  = 30,     (8)\r\n" );
document.write( " +  = 34,     (9)\r\n" );
document.write( " +  = 37.     (10)\r\n" );
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document.write( "Add all ten equations (1) - (10) at once (both left and right sides). You will get an equation\r\n" );
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document.write( " = ,    or,  after dividing both sides by 4\r\n" );
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document.write( " = .    (*)\r\n" );
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document.write( "Now distract, one by one and one after one, the above equations (10), (9), (8), (7) and (3) from the equation (*) as it is shown below. \r\n" );
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document.write( " = 29,     (11)    ( distract (10) from (*);   29 = 66 - 37 )\r\n" );
document.write( " = 32,     (12)    ( distract  (9) from (*);   32 = 66 - 34 )\r\n" );
document.write( " = 36,     (13)    ( distract  (8) from (*);   36 = 66 - 30 )\r\n" );
document.write( " = 37,     (14)    ( distract  (7) from (*);   37 = 66 - 29 )\r\n" );
document.write( " = 44,     (15)    ( distract  (3) from (*);   44 = 66 - 22 )\r\n" );
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document.write( "Now,  compare (1) and (11). You will get   = 29 - 16 = 13.   (16)\r\n" );
document.write( "Next, compare (1) and (12). You will get   = 32 - 16 = 16.   (17)\r\n" );
document.write( "      Compare (1) and (13). You will get   = 36 - 16 = 20.   (18)\r\n" );
document.write( "      Compare (8) and (14). You will get   = 37 - 30 =  7.   (19)\r\n" );
document.write( "      Compare (9) and (15). You will get   = 44 - 34 = 10.   (20)\r\n" );
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document.write( "Now, to answer the question, multiply the right sides of (16) - (20). You will get\r\n" );
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document.write( " = 13*16*20*7*10 = 291200.\r\n" );
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document.write( "Answer.  The product of five numbers is 291200.\r\n" );
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