document.write( "Question 1197061: There are 1859 balls inside an urn from 1 to 1859. A ball is selected at random. Given that the selected ball has a number that is not relatively prime to 1859, what is the probability that it is divisible by 13 only? \n" ); document.write( "

Algebra.Com's Answer #830142 by ikleyn(52898) $\"\"$ $\"About$

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\n" ); document.write( "There are 1859 balls inside an urn from 1 to 1859. A ball is selected at random.
\n" ); document.write( "Given that the selected ball has a number that is not relatively prime to 1859,
\n" ); document.write( "what is the probability that it is divisible by 13 only?
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document.write( "The prime decomposition of the number 1859 is  \r\n" );
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document.write( "          1859 = .      (1)\r\n" );
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document.write( "    To find this prime decomposition quickly and without tortures,\r\n" );
document.write( "    ask GOOGLE \"primary decomposition of the number 1859\".\r\n" );
document.write( "    You will get web-sites with online solvers, which provide the answer to you in one second. \r\n" );
document.write( "    Concretely, this web-site gives the answer https://www.factors-of.com/Prime-decomposition-of_1859_\r\n" );
document.write( "    Alternatively, you can find prime divisors manually using sieve of Eratosthenes.\r\n" );
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document.write( "You are given that randomly selected ball has a number that is not relatively prime to 1859.\r\n" );
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document.write( "It means that this randomly selected number is divisible by 11 or by 13 (or both).\r\n" );
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document.write( "The subset N(11) of integers from  {1,2,3, . . .  ,1859}  that are divisible by 11, has  1859/11 = 169 elements.\r\n" );
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document.write( "The subset N(13) of integers from  {1,2,3, . . .  ,1859}  that are divisible by 13, has  1859/13 =  143 elements.\r\n" );
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document.write( "The subset N(11*13) of integers from  {1,2,3, . . .  ,1859}  that are divisible by both 11 and 13, has  1859/(11*13) =  13 elements.\r\n" );
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document.write( "    (this subset is the intersection of N(11) and N(13), since 11 and 13 are relatively prime).\r\n" );
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document.write( "The subset N(11,13) of integers from  {1,2,3, . . . ,1859}  that are not relatively prime to 1859 is the union of N(11) and N(13)\r\n" );
document.write( "and has  N(11) + N(13) - N(11*13) = 169 + 143 - 13 = 299 elements.\r\n" );
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document.write( "    It is the \"given\" subset, to which the selected number \r\n" );
document.write( "    does belong, as it is given in the condition.\r\n" );
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document.write( "The subset of N(11,13) which consists of integers divisible by 13 only is N(13) \ N(11*13), and it has 143 - 13 = 130 elements.\r\n" );
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document.write( "The probability under the problem's question is this ratio  P =  = .\r\n" );
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