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Each visitor at the Wacky World Water Park gets a free towel to use at the park. At the entrance, workers arrange the towels in stacks with the same number of towels in each stack. Each stack must have more than 1 towel and there must be more than 1 stack of towels. Which of these numbers of towels could be arranged into more than 1 stack with the same number of towels in each stack? Each stack must have more than 1 towel. Explain why this arrangement is possible or not possible for each of these numbers of towels. 60 29 37 42

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frika

Consider these numbers in turn.

1. 60. This number is composite, because 60=2·2·3·5. Acoording to the rule given in task for this number you can have such possibilities:

  • 2 stacks with 30 towels at each;
  • 3 stacks with 20 towels at each;
  • 4 stacks with 15 towels at each;
  • 5 stacks with 12 towels at each;
  • 6 stacks with 10 towels at each;
  • 10 stacks with 6 towels at each;
  • 12 stacks with 5 towels at each;
  • 15 stacks with 4 towels at each;
  • 20 stacks with 3 towels at each;
  • 30 stacks with 2 towels at each.

2. 29 is prime number, because 29=1·29 (has only two trivial divisors). Then you cannot choose numbers of stacks and towels according to the given rule.

3. 37 is prime number, because 37=1·37 (has only two trivial divisors). Then you cannot choose numbers of stacks and towels according to the given rule.

4. 42=2·3·7 is composite number. Acoording to the rule given in task for this number you can have such possibilities:

  • 2 stacks with 21 towels at each;
  • 3 stacks with 14 towels at each;
  • 6 stacks with 7 towels at each;
  • 7 stacks with 6 towels at each;
  • 14 stacks with 3 towels at each;
  • 21 stacks with 2 towels at each.

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