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Probability and divisibility by 30

Well, probability is on my mind - this sounds like a great song title, no?

I've got another probability problem, this one presented to me by my brother Abe as I called him from the airport half way to my awesome vacation in Hawaii. I chewed on it for a little while then called him back with the answer, got his answering machine and left the answer there to impress him. He didn't call me back but I think I got the answer right. Maybe that's why he didn't call?

Here's the problem:

What is the probability that a 10 digit number will be divisible by 30?

The problem was really stated as asking how many ten digit numbers are that are divisible by 30, which is in essence the same problem.

This is a neat problem in that it involves several aspects of solving Math problems and has a number of dimensions to it that lead to a pleasant exploration. Some ideas on approaching this problem:

  1. Once must assume that a 10 digit number has no leading zeros, i.e. that the first digit is between 1 and 9. How does this special case for the first digit affect the probability? Without this consideration someone might hastily and incorrectly conclude that the probability is real close to 999,999,999 / 30.
  2. How many 10 digit numbers are there? How would you count them?
  3. If you take the number of 10 digit numbers and divide that number by 30 do you then get the right answer? Why or why not?
  4. What is interesting about the number 30? How would you test for divisibility by 30? Hint: Factor 30 into primes.
  5. Does the fact that 30 ends in zero lead to a simplification of the problem?
  6. Can you solve this problem for numbers with fewer digits? How many single digit numbers are there divisible by 30? How many 2 digit numbers? How many such 3, 4, and 5 digit numbers are there?
  7. What if the problem asked about divisibility by some other number, maybe one that fewer or more factors? Might it then be easier to utilize an approach that didn't take into consideration the factors of the number we're testing divisibility for?
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  1. Calculating numerator and denominator separately is important, but in this case starting with the denominator may help with the numerator.

    There are 9,999,999,999 numbers of no more than 10 digits, but of those 999,999,999 are of no more than 9 digits, leaving 9,000,000,000 ten-digit numbers.

    This suggests (maybe) a nice way of counting the correct numerator.

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