document.write( "Question 1074390: If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 4133.\r
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Algebra.Com's Answer #689055 by Edwin McCravy(20056)\"\" \"About 
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document.write( "If we find the sum of all the multiples of 3 below 4133,\r\n" );
document.write( "and add that to the sum of all the multiples of 5 below 4133,\r\n" );
document.write( "we will have added the multiples of 15 below 4133 twice.  So \r\n" );
document.write( "then if we subtract the sum of all the multiples of 15 from\r\n" );
document.write( "that we'll have the desired sum.\r\n" );
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document.write( "We find the sum of the multiples of 3 below 4133:\r\n" );
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document.write( "The least multiple of 3 below 4133 is 3.\r\n" );
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document.write( "4133/3 = 1377.67 so the largest multiple of 3 less that 4133\r\n" );
document.write( "is 1377*3 or 4131. \r\n" );
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document.write( "We want the sum of the arithmetic sequence:  \r\n" );
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document.write( "3,6,9,...,4131\r\n" );
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document.write( "We use the sum formula\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"\r\n" );
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document.write( "We need a1=3 and an=4131\r\n" );
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document.write( "It's obvious to me that the number of terms is n = 1377, but if\r\n" );
document.write( "it's not obvious to you, just divide them all by 3 and\r\n" );
document.write( "get the sequence\r\n" );
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document.write( "1,2,3,...,1377\r\n" );
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document.write( "and now you can see that indeed n = 1377.\r\n" );
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document.write( "We substitute in the sum formula\r\n" );
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document.write( "\"S%5B1377%5D=expr%281377%2F2%29%283%2B4131%29=2846259\"\r\n" );
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document.write( "We find the sum of the multiples of 5 below 4133:\r\n" );
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document.write( "The least multiple of 5 below 4133 is 5.\r\n" );
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document.write( "4133/5 = 826.6 so the largest multiple of 3 less that 4133\r\n" );
document.write( "is 826*5 or 4130. \r\n" );
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document.write( "We want the sum of the arithmetic sequence:  \r\n" );
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document.write( "5,10,15,...,4130\r\n" );
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document.write( "We use the sum formula again:\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"\r\n" );
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document.write( "We need a1=5 and an=4130\r\n" );
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document.write( "As before, it's obvious to me that the number of terms is \r\n" );
document.write( "n = 826, but again, if it's not obvious to you, just divide \r\n" );
document.write( "them all by 5 and get the sequence\r\n" );
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document.write( "1,2,3,...,826\r\n" );
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document.write( "and now you can see that indeed n=826.\r\n" );
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document.write( "We substitute in the sum formula\r\n" );
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document.write( "\"S%5B826%5D=expr%28826%2F2%29%285%2B4130%29=1707755\"\r\n" );
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document.write( "We find the sum of the multiples of 15 below 4133:\r\n" );
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document.write( "The least multiple of 15 below 4133 is 15.\r\n" );
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document.write( "4133/15 = 275.33 so the largest multiple of 15 less that 4133\r\n" );
document.write( "is 275*15 or 4125. \r\n" );
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document.write( "We want the sum of the arithmetic sequence:  \r\n" );
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document.write( "15,30,45,...,4125\r\n" );
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document.write( "We use the sum formula\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"\r\n" );
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document.write( "We need a1=15 and an=4125\r\n" );
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document.write( "Once more, it's obvious to me that the number of terms is \r\n" );
document.write( "n = 275, but if it's not obvious to you, just divide them all \r\n" );
document.write( "by 15 and get the sequence\r\n" );
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document.write( "1,2,3,...,275\r\n" );
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document.write( "and now you can see that indeed n=275.\r\n" );
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document.write( "We substitute in the sum formula\r\n" );
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document.write( "\"S%5B275%5D=expr%28275%2F2%29%2815%2B275%29=569250\"\r\n" );
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document.write( "So we add the sum of the multiples of 3 and the sum of the\r\n" );
document.write( "multiples of 5, and get 2846259+1707755 = 4554014\r\n" );
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document.write( "That adds the multiples of 15 twice, since multiples of 15\r\n" );
document.write( "are multiples of both 3 and 5.\r\n" );
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document.write( "So we subtract the sum of the multiples of 15 and get\r\n" );
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document.write( "4554014-569250 = 3984764  <-- answer\r\n" );
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document.write( "Edwin

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