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.
\n" ); document.write( "a) How many balls must be printed to get the first ball with stripes, polka dots, and stars?
\n" ); document.write( "b) How many balls have been printed at that point that have only two of the designs?
\n" ); document.write( "

Algebra.Com's Answer #633108 by KMST(5328)\"\" \"About 
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 \"red%2818%29\" , \"blue%2815%29\" , and \"green%2830%29\" .
\n" ); document.write( "That number is \"highlight%2890%29=3%2Agreen%2830%29=6%2Ablue%2815%29=5%2Ared%2818%29\" .
\n" ); document.write( "The \"highlight%2890th%29\" ball has all 3 designs.
\n" ); document.write( "How do we find it? Let me count the ways.
\n" ); document.write( "1) We could start checking increasing multiples of \"green%2830%29\" until we find one that is also a multiple of \"red%2818%29\" , and \"blue%2815%29\" .
\n" ); document.write( "\"1%2Agreen%2830%29=2%2Ablue%2815%29\" is already a multiple of \"blue%2815%29\" ,
\n" ); document.write( "so all multiples of \"green%2830%29\" will be multiple of \"blue%2815%29\" ,
\n" ); document.write( "but \"1%2Agreen%2830%29\" is not a multiple of \"red%2818%29\" ,
\n" ); document.write( "and neither is \"2%2Agreen%2830%29=60\" ,
\n" ); document.write( "but \"3%2Agreen%2830%29=5%2Ared%2818%29=highlight%2890%29\" is the first common multiple of \"red%2818%29\" , \"blue%2815%29\" , and \"green%2830%29\" that we find.
\n" ); document.write( "2) We could start from the prime factorizations:
\n" ); document.write( "\"red%2818%29=2%2A9=2%2A3%5E2\" ,
\n" ); document.write( "\"blue%2815%29=3%2A5\" , and
\n" ); document.write( "\"green%2830%29=3%2A10=2%2A5%2A3\" ,
\n" ); document.write( "so the least common multiple must contain the prime factors \"2\" , \"3\" , and \"5\" ,
\n" ); document.write( "and we need the \"3\" to be squared, as it is in \"red%2818%29=2%2A3%5E2\" ,
\n" ); document.write( "so the least common multiple is \"2%2A3%5E2%2A5=2%2A9%2A5=10%2A9=highlight%2890%29\" .
\n" ); document.write( "
\n" ); document.write( "b) We already saw in 1) above that all multiples of \"green%2830%29\" are multiples of \"blue%2815%29\" ,
\n" ); document.write( "so the balls number \"green%2830%29=2%2Ablue%2815%29\" and \"60=2%2Agreen%2830%29=4%2Ablue%2815%29\" have two designs:
\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 \"highlight%282%29\" balls, the 30th and the 60th one, will have been printed with only two designs:
\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 \"red%2818%29=2%2A9=2%2A3%5E2\" , and \"blue%2815%29=3%2A5\" ,
\n" ); document.write( "or \"red%2818%29=2%2A9=2%2A3%5E2\" , and \"green%2830%29=3%2A10=2%2A5%2A3\" ,
\n" ); document.write( "must be a multiple of all three numbers.
\n" ); document.write( "It must have \"2\" , \"3%5E2\" , and \"5\" in its factorization,
\n" ); document.write( "so it will be a multiple of \"2%2A3%5E2%2A5=90\" .
\n" ); document.write( "If a ball has stripes and one another design printed, it has all three designs.
\n" ); document.write( "
\n" );