document.write( "Question 891345: In the following equation, A, B, C, and D are all positive integers such that A has 4 factors, B has 6 factors, and D has 8 factors. What is the smallest possible sum of A, B, C, and D, if (A×B)/C = D ? \n" ); document.write( "
Algebra.Com's Answer #539697 by Edwin McCravy(20056)\"\" \"About 
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document.write( "positive integers of the form pq, and only those of the form pq,\r\n" );
document.write( "have 4 factors, 1,p,q, and pq  where p and q are different primes.\r\n" );
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document.write( "So let A = pq\r\n" );
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document.write( "Positive integers of the form rs², and only those of the form rs²,\r\n" );
document.write( "have 6 factors, 1,r,s,s²,rs and rs²  where r and s are different primes.\r\n" );
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document.write( "So let B = ra²\r\n" );
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document.write( "Positive integers of the form tuv, and only those of the form tuv,\r\n" );
document.write( "have 8 factors, 1,t,u,v,tu,tv,uv and tuv  where t, u and v are \r\n" );
document.write( "3 different primes.\r\n" );
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document.write( "So let D = tuv\r\n" );
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document.write( "Since D = AB/C, C = AB/D = pqrs²/(tuv)\r\n" );
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document.write( "So we have \r\n" );
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document.write( "A = pq  we can assume p < q\r\n" );
document.write( "B = ra²\r\n" );
document.write( "C = pqrs²/(tuv)\r\n" );
document.write( "D = tuv we can assume t < u < v\r\n" );
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document.write( "We will try to use only the smallest three primes, {2,3,5} if possible.\r\n" );
document.write( "If so, then A and B can be made smaller than D which is the product\r\n" );
document.write( "of three different primes.\r\n" );
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document.write( "We make D as small as possible by choosing t=2, u=3, v=5, so D=30.\r\n" );
document.write( "We make B as small as possible by choosing r=3, s=2, so B=3×2² = 12.\r\n" );
document.write( "We would like to make A as small as possible, which would be 6 with p=2 and q=3.\r\n" );
document.write( "But that won't do, because since v=5, C = pqrs²/(tuv), then pqrs² must\r\n" );
document.write( "be divisible by 5, and since neither r nor s are 5, the larger of p,q \r\n" );
document.write( "must be 5, so q=5, and we can choose p=2.  So A = pq = 2×5 = 10, and\r\n" );
document.write( "C = pqrs²/(tuv) = 2×5×3×2²/(2×3×5) = 4\r\n" );
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document.write( "So A+B+C+D = 10+12+4+30 = 56, the smallest possible sum.\r\n" );
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document.write( "Edwin
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