### 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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## Comments

9 ways to arrange the digits 1-9 to form a 9-digit number?

I'm confused. If my calculations are correct, wouldn't there be 362880 (9*8*7*6*5*4*3*2*1) possible ways to arrange 1-9 into a nine digit number? This doesn't change the answer but confusing still.

I'm confused. If my calculations are correct, wouldn't there be 362880 (9*8*7*6*5*4*3*2*1) possible ways to arrange 1-9 into a nine digit number? This doesn't change the answer but confusing still.

Scubee, 9! means 9 factorial which is 362, 880.

Actually, there would be 3,486,784,401 ways

Congratulations on having the first approved teaser with a 4 in the ten-thousands position.

Firstly, thanks calmsavior! And secondly, where did your idea come from Holo?

Wow. I would have never figured that out.... I guess I'm not as smart as you guys... Very clever!

One problem. 9! means 9*8*7*6*5*4*3*2*1, not 9+8+7+6+5+4+3+2+1.

Nevermind my last comment. I understand where you got the 9+8+7+6+5+4+3+2+1 from now.

I must say this one incorporates a simple rule and disguises it very well; good job!

Whoo. This teaser was amazing. It actually took me a minute...

But then I got it. (With the hint... )

But then I got it. (With the hint... )

I like the way this simple problem disguises such a simple division rule. It took me a while to notice it. I must boast that I didn't have to look at the hint though

Too easy.

Which numbers are prime? Oh look, 1+...+9 = 45 which is divisible by 3 and you're done.

Which numbers are prime? Oh look, 1+...+9 = 45 which is divisible by 3 and you're done.

This problem seems incorrect.. What about #'s such as 127, 137, 149, 157 ... etc?

http://en.wikipedia.org/wiki/List_of_prime_numbers

http://en.wikipedia.org/wiki/List_of_prime_numbers

Oops never mind. I thought this problem was for 3 digits.

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