January 2017 Brain Puzzler Solution

Q:  The digits one through nine are randomly arranged to make a 9 digit number.
What is the probability that the resulting number is divisible by eighteen?

A:  The number will always be divisible by 9 as the digits 1 – 9 add up to 45 (any number whose digits add up 9 or a multiple of 9 will be divisible by 9).    So every one of these 9 digit numbers will be divisible by 9.
If the number is also divisible by 2, then it will be divisible by 18.
So we need to find how often the number will be divisible by 2.
The last digit could be any number from 1-9 and 4 of those digits are even (2, 4, 6, 8) making the # divisible by 2 (and 9, see above).
So there is a 4/9 probability that the number will be divisible by 18.

BONUS: What is the probability that the resulting number is divisible by 27?
Still working on the divisibility by 27…please send comments

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