### Brain Teasers

# Prime Nine

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

There are 9! ways to arrange the digits 1-9 to form a 9-digit number.

E.g. 345, 987, 126

If you know this, then if I wrote all 9! possibilities on cards (1 way per card), and I asked you to select one at random; what is the probability that the number you selected is prime?

E.g. 345, 987, 126

If you know this, then if I wrote all 9! possibilities on cards (1 way per card), and I asked you to select one at random; what is the probability that the number you selected is prime?

### Hint

Divisibility tricks are useful now.### Answer

Zero.A well known divisibility rule is that if a number's digits add to 9, the number itself is a multiple of 9.

(E.g. 18, 1+8=9, 18 is a multiple of 9

24786, 2+4+7+8+6=27, 2+7=9, 24786 is a multiple of 9)

Given the fact that 1+2+...+8+9=45, which is a multiple of 9; this means that all 9! numbers are multiples of 9.

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