Puzzle #28: Mystery Number

Consider the natural numbers from 1 to 200.

Consider the following properties that any one such number (n) may have:

  • n is not even.
  • n is prime.
  • n is evenly divisible by 3.
  • n is evenly divisible by 4.
  • n is evenly divisible by 5.
  • n is evenly divisible by 7.
  • n is evenly divisible by 11.

Find the values for which the maximum number of the above properties is true. It need not be the same set of properties if there is more than one value.

(Hint: Do not use brute force. Consider how the properties interrelate. For example, if n is evenly divisible by 4, it can not be not even, and it can not be prime.)

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, December 18, 2013 at noon Pacific Time. I will post the answer shortly after.