document.write( "Question 977539: Hi there tutors, can you help me out answer this question? Thanks\r
\n" ); document.write( "\n" ); document.write( "Five consecutive positive integers, p, q, r, s and t, each less than 10000, produce a sum which is a perfect square, while the sum q + r + s is a perfect cube. What is the value of \"+sqrt%28+p+%2B+q+%2B+r+%2B+s+%2B+t+%29+\"? \n" ); document.write( "
Algebra.Com's Answer #599100 by AnlytcPhil(1806)\"\" \"About 
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Five consecutive positive integers, p, q, r, s and t, each less than 10000,
\n" ); document.write( "produce a sum which is a perfect square, while the sum q + r + s is a perfect
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document.write( "We have consecutive positive integers p, q=p+1, r=p+2, s=p+3, t=p+4\r\n" );
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document.write( "p+q+r+s+t = A², where A is some positive integer\r\n" );
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document.write( "p+(p+1)+(p+2)+(p+3)+(p+4) = 5p+10 = A²\r\n" );
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document.write( "5p+10 = A²\r\n" );
document.write( "5(p+2) = A²\r\n" );
document.write( "So A is a multiple of 5, say A = 5B where B is a positive integer.\r\n" );
document.write( "5(p+2) = (5B)²\r\n" );
document.write( "5(p+2) = 25B²\r\n" );
document.write( "p+2 = 5B²\r\n" );
document.write( "p = 5B²-2\r\n" );
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document.write( "q+r+s = C³ where C is a positive integer\r\n" );
document.write( "(p+1)+(p+2)+(p+3) = C³\r\n" );
document.write( "3p+6 = C³\r\n" );
document.write( "3(p+2) = C³\r\n" );
document.write( "So C is a multiple of C, say C = 3D where D is a positive integer.\r\n" );
document.write( "3(p+2) = (3D)³\r\n" );
document.write( "3(p+2) = 27D³\r\n" );
document.write( "p+2 = 9D³\r\n" );
document.write( "p = 9D³-2\r\n" );
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document.write( "p = 5B²-2 = 9D³-2\r\n" );
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document.write( "p+2 = 5B² = 9D³\r\n" );
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document.write( "B must be a multiple of 3 to create the 9 factor on the right.\r\n" );
document.write( "D must be a multiple of 5 to create the 5 factor on the left\r\n" );
document.write( "Then B must be a multiple of 5 since D³ will produce 3 factors of 5.\r\n" );
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document.write( "So B must be a multiple of 3 and 5, so we take B = 15 and D = 5\r\n" );
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document.write( "p+2 = 5(15)² = 9(5)³\r\n" );
document.write( "p+2 = 1125 = 1125\r\n" );
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document.write( "That works, so\r\n" );
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document.write( "So p = 1125-2 = 1123\r\n" );
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document.write( "So (p,q,r,s,t) = (1123,1124,1125,1126,1126)\r\n" );
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document.write( "Checking\r\n" );
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document.write( "p+q+r+s+t = 1123+1124+1125+1126+1127 = 5625 = 75²\r\n" );
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document.write( "q+r+s = 3375 = 15³\r\n" );
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What is the value of \"+sqrt%28+p+%2B+q+%2B+r+%2B+s+%2B+t+%29+\"?
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document.write( "Answer: \"sqrt%28p%2Bq%2Br%2Bs%2Bt%29\"\"%22%22=%22%22\"\"sqrt%285625%29\"\"%22%22=%22%22\"\"75\" \r\n" );
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document.write( "Edwin
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