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Call a number a digital number if it consists of all the digits from 1-9, each used exactly once. What is the probability that a digital number will be divisible by 7 ? What is the probability that a digital number will be of the form 9p, where p is a prime?

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    What haven't you tried?2012-07-04
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    @Gigili: I just LOVE your question :-)2012-07-04
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    I have no idea where to start! At least on the first part, we can BEGIN to use the divisibility rule by 7, but even then it gets too cumbersome to do. As far as the second part (9p) i have absolutely no clue where to start!2012-07-04
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    My approach would be to write a program to find all the pandigital numbers divisible by 7 and all those of the form 9p. Shouldn't be too hard.2012-07-04
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    I don't know how or why, but I have a hunch that the fact that $1001$ is divisible by $7$ will turn out to be useful. This implies that $$1000000a+1000b+c\equiv a-b+c\pmod7.$$ This means that for a digital number to be divisible by seven, we must have the equation $b=a+c\pmod 7$, where $a,b,c$ are the integers formed by groups of three digits, starting from the most significant.2012-07-04
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    I agree it would be simple to make a program to solve this, but perhaps is there a more elegant solution requiring mathematical concepts?2012-07-04
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    ...What divisibility rule by 7?2012-07-04
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    http://www.maa.org/mathland/mathtrek_05_23_05.html2012-07-04
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    @Yamini There is no chance of an elegant solution to the $9p$ problem, unless the probability is $0$. And since $123458679$ is nine times a prime, the probability is non-zero.2012-07-04
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    Ouch! Erick's discovery is bad news, indeed. What course is this homework for anyway? If programming, then...2012-07-04
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    Well, it was a problem I found on a Stanford Math Circle homework packet a while back, and I just found it from when i posted it on Art of Problem Solving (it remained unsolved there for a year). I will try to find it again. Because it was on a Math Circle homework, i suspected it would have a pure mathematical solution2012-07-04
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    Here it is, problem #2. This is why i suspect it has a simple solution. http://math.stanford.edu/circle/notes05f/cong.pdf2012-07-04
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    Thanks, Yamini. That sounds a lot better.2012-07-04
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    Matlab results for the first problem: out of 362880 digital numbers, 51752 are divisible by 7.2012-07-04
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    Matlab results for the second problem: out of 362880 digital numbers, 23895 factor as 9*p for a prime number p.2012-07-04
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    The funny thing about digital (or pandigital) numbers is that they are all multiples of 9.2012-08-30

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