document.write( "Question 1009054: If the product represented by 274! Is divisible by 12 to the power
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Algebra.Com's Answer #624697 by Edwin McCravy(20056)\"\" \"About 
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If the product represented by 274! Is divisible by 12 to the power
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document.write( "n will be the number of factors of 12 there are in 274!\r\n" );
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document.write( "12 = 2×2×3\r\n" );
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document.write( "Each factor of 12 in 274! amounts to two factors of 2 and \r\n" );
document.write( "one factor of 3.\r\n" );
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document.write( "We are interested in how many factors of 2 and how many \r\n" );
document.write( "factors of 3 are contained in\r\n" );
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document.write( "a)   274! = 1×2×3×4×...×274\r\n" );
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document.write( "Let's first find out how many factors of 3 are contained in 274!\r\n" );
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document.write( "Product a) contains \r\n" );
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document.write( " 91 multiples of 3, since 274/3 = 91.333...\r\n" );
document.write( " 30 multiples of 3^2, or 9, since 274/9 = 30.444...\r\n" );
document.write( " 10 multiples of 3^3, or 27, since 274/27 = 10.148...\r\n" );
document.write( "  3 multiples of 3^4, or 81, since 274/81 = 3.3827... \r\n" );
document.write( "  1 multiple of 3^5, or 243, since 274/243 = 1.12767...\r\n" );
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document.write( "135 factors of 3 contained in 274!\r\n" );
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document.write( "Since every factor of 12 contained in 274! has exactly 1\r\n" );
document.write( "factor of 3, 135 will be the number of factors of 12 in \r\n" );
document.write( "274!, provided that there are at least twice that many\r\n" );
document.write( "factors of 2 in 274!, since each factor of 12 amounts to \r\n" );
document.write( "2 factors of 2 and 1 factor of 12.  So we must make sure \r\n" );
document.write( "that 274! contains at least twice 135 or 270 factors of \r\n" );
document.write( "2 in order to claim that it has 135 factors of 12.  So \r\n" );
document.write( "let's find out if there are enough factors of 2 to \r\n" );
document.write( "justify that 135 is the correct answer.\r\n" );
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document.write( "a)   274! = 1×2×3×4×...×274\r\n" );
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document.write( "Product a) contains \r\n" );
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document.write( "137 multiples of 2, since 274/2 = 137\r\n" );
document.write( " 68 multiples of 2^2, or 4, since 274/4 = 68.5\r\n" );
document.write( " 34 multiples of 2^3, or 8, since 274/8 = 34.25\r\n" );
document.write( " 17 multiples of 2^4, or 16, since 274/16 = 17.125 \r\n" );
document.write( "  8 multiples of 2^5, or 32, since 274/32 = 8.5625\r\n" );
document.write( "  4 multiples of 2^6, or 64, since 274/64 = 4.28125\r\n" );
document.write( "  2 multiples of 2^7, or 128, since 274/128 = 2.140625\r\n" );
document.write( "  1 multiple of 2^8, or 256, since 274/256 = 1.0703125\r\n" );
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document.write( "271 factors of 2 contained in 274!\r\n" );
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document.write( "So there is just 1 extra factor of 2 than 270, the number \r\n" );
document.write( "necessary to make there be 135 factors of 12 in 274!\r\n" );
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document.write( "Answer: 274! contains 135 factors of 12 Therefore n = 135 \r\n" );
document.write( "is the largest possible value of n. \r\n" );
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document.write( "Edwin
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