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September 26th, 2013, 10:55 AM   #1
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HELP! probability that a 7 digit number is divisible by 7?

The seven digits {1; 2; 3; 4; 5; 6; 7} are written down in a random order. What is the
chance that the resulting number is divisible by 7? For instance, 1234576 works, but
1234567 doesn’t.
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September 26th, 2013, 11:48 AM   #2
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Re: HELP! probability that a 7 digit number is divisible by

The answer to the question is the number of numbers with digits 1,2,3,4,5,6,7 that are divisible by 7 divided by 7!.

I don't know how you determine all such numbers divisible by 7 but here are a few:

1427356
6427351
1357426
6357421
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September 26th, 2013, 12:33 PM   #3
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Re: HELP! probability that a 7 digit number is divisible by

Quote:
Originally Posted by icemanfan
The answer to the question is the number of numbers with digits 1,2,3,4,5,6,7 that are divisible by 7 divided by 7!.

I don't know how you determine all such numbers divisible by 7 but here are a few:

1427356
6427351
1357426
6357421
Your first example is interesting to me cause 7 acts as a 'symmetric divider' (I just made up that term )

14 27356
1 42 7356
1427 35 6
14273 56

The numbers in bold are all divisible by 7

So must be 27356 and 14273 (For obvious reason)

Same with your last example

63 57421
6 35 7421
6357 42 1
63574 21

The numbers in bold are all divisible by 7

So must be 57421 and 63574 (For obvious reason)

I want to add

1234765
1276534

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September 26th, 2013, 07:53 PM   #4
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Re: HELP! probability that a 7 digit number is divisible by

There's 720 of those critters:
1: 1234576
2: 1234765
...
719: 7653142
720: 7654213

720/5040 = 1/7
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September 26th, 2013, 08:41 PM   #5
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Re: HELP! probability that a 7 digit number is divisible by

Yes, and the first seven appear as the 2nd, 6th, 9th, 37th, 41st, 44th and 50th members of the sequence of ascending numbers that have the unrepeated digits 1, 2, 3, 4, 5, 6 and 7. Is it mere coincidence that 6! = 720?
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September 27th, 2013, 10:37 AM   #6
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Re: HELP! probability that a 7 digit number is divisible by

Quote:
Originally Posted by Denis
There's 720 of those critters:
1: 1234576
2: 1234765
...
719: 7653142
720: 7654213

720/5040 = 1/7
Hey "Denis" How did you get all these numbers?
Can you please tell me?
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September 27th, 2013, 07:22 PM   #7
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Re: HELP! probability that a 7 digit number is divisible by

In a tremendously shrewd, clever and highly advanced manipulation:

loop a,b,c,d,e,f,g from 1 to 7 keeping 'em all different
n = a*10^6 + b*10^5 + c*10^4 + d*10^3 + e*10^2 + f*10 + g
if n@7 = 0 then count = count + 1 : print n

Would you like my autograph?
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September 30th, 2013, 01:51 AM   #8
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Re: HELP! probability that a 7 digit number is divisible by

If you look at the value of 10^n (mod 7), you see that for any permutation of the digits 1-7, the remainder mod (7) is:

a + b + 2c + 3d + 4e + 5f + 6g

Where a-g is a permutaion of the digits 1-7.

Note that the digits here are not in the same order as they appear in the decimal expansion.

It appears that if you keep the digits in the same order, but cycle them round, then you get each value 0-6 (mod 7) precisely once. This would explain why you get 6! numbers divisible by 7 and would also imply that you get 6! numbers that are n (mod 7) n = 0-6.

I can't see how to prove this.
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October 3rd, 2013, 01:58 AM   #9
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Re: HELP! probability that a 7 digit number is divisible by

Quote:
Originally Posted by lincoln40113
How did you get all these numbers?
This page describes how you can find them.
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October 4th, 2013, 02:10 PM   #10
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Re: HELP! probability that a 7 digit number is divisible by

I believe that everyone may have been "overthinking" this one.

The probability that a random number is divisible by 7 is obviously 1 in 7. This is just due to the fact that every seventh number is divisible by 7 (starting at 7, 14, 21....).
So given a set of random numbers, a sequential set of numbers, or in this case 7! (5,040) numbers... The randomness applies and does not skew the probability of 1/7.

Just another thought on the topic for what it is worth.

I also wonder how "lincoln40113" was able to determine 720 favorable outcomes. Was this brute force? and if so... cudos on that effort!!!
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