document.write( "Question 1016707: Balls are printed in a pattern. Every 18th ball has stripes, every 15th ball has polka dots, and every 30th ball has stars.
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document.write( "a) How many balls must be printed to get the first ball with stripes, polka dots, and stars?
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document.write( "b) How many balls have been printed at that point that have only two of the designs? \n" );
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Algebra.Com's Answer #633108 by KMST(5328)![]() ![]() You can put this solution on YOUR website! a) The ordinal number of the first ball with all 3 designs is the least common multiple of \n" ); document.write( "That number is \n" ); document.write( "The \n" ); document.write( "How do we find it? Let me count the ways. \n" ); document.write( "1) We could start checking increasing multiples of \n" ); document.write( " \n" ); document.write( "so all multiples of \n" ); document.write( "but \n" ); document.write( "and neither is \n" ); document.write( "but \n" ); document.write( "2) We could start from the prime factorizations: \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "so the least common multiple must contain the prime factors \n" ); document.write( "and we need the \n" ); document.write( "so the least common multiple is \n" ); document.write( " \n" ); document.write( "b) We already saw in 1) above that all multiples of \n" ); document.write( "so the balls number \n" ); document.write( "polka dots, and stars, but not stripes. \n" ); document.write( "So by the time the first ball with all three designs, the 90th ball, is printed, \n" ); document.write( "at least \n" ); document.write( "polka dots, and stars, but not stripes. \n" ); document.write( "Are there any other balls before the 90th ball that have two designs? \n" ); document.write( "Maybe stripes and polka dots, or stripes and stars? \n" ); document.write( "NO, there is none. \n" ); document.write( "Any multiple of \n" ); document.write( "or \n" ); document.write( "must be a multiple of all three numbers. \n" ); document.write( "It must have \n" ); document.write( "so it will be a multiple of \n" ); document.write( "If a ball has stripes and one another design printed, it has all three designs. \n" ); document.write( " |