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Numbers with nothing in the middle

There is an infinite set of numbers with nothing in the middle.

By that I mean a positive whole number whose first digit and last digit are not zero, but any and all digits in between are zero.

The 81 numbers between 11 and 99 that are not multiples of ten are numbers with nothing in the middle. So are the 81 numbers of form x0y between 101 and 909. So are the 81 numbers of form x00y between 1001 and 9009. And so on.

I started wondering, which of these numbers is divisible by each prime number? What is the lowest prime number whose multiples include a complete set of numbers with nothing in the middle?

2 is a divisor of all even numbers with nothing in the middle:

12 22 32 42 52 62 72 82 92
14 24 34 44 54 64 74 84 94
16 26 36 46 56 66 76 86 96
18 28 38 48 58 68 78 88 98

and likewise for 102, 104, etc; but obviously 2 is not a divisor of odd numbers with nothing in the middle.

3 is a divisor of all numbers with nothing in the middle whose digits add up to 3:

21 51 81
12 42 72
33 63 93
24 54 84
15 45 75
36 66 96
27 57 87
18 48 78
39 69 99

and likewise for 201, etc; but obviously 3 is not a divisor of numbers with nothing in the middle whose digits to not add up to a multiple of 3.

5 is a divisor of all numbers with nothing in the middle whose last digit is 5,only nine possible configurations:

15 25 35 45 55 65 75 85 95

and so on for 105, 205, etc.

7 is a divisor of a set of numbers with nothing in the middle, which includes all first-digit/last-digit combinations except where only one of the two is 7.

14 21 35 42 56 63 77 84 91
28 49 98
105 203 301 406 504 602 805 903
308 609
1,001 2,002 3,003 4,004 5,005 6,006 8,001 9,002
1,008 2,009 8,008 9,009
10,003 20,006 30,002 40,005 50,001 60,004 80,003 90,006
30,009 50,008
100,002 200,004 300,006 400,001 500,003 600,005 800,002 900,004
100,009 400,008 800,009
1,000,006 2,000,005 3,000,004 4,000,003 5,000,002 6,000,001 8,000,006 9,000,005
5,000,009 6,000,008

And so on with 10,000,004, etc. So that's 65 of the possible 81 combinations covered. Can we do better?

11 is a divisor of the set of numbers with nothing in the middle which 1) have an even total number of digits, and the first and last are the same; or 2) have an odd number of digits, and the first and last add up to 11.

11 22 33 44 55 66 77 88 99
209 308 407 506 605 704 803 902

and then 1001, 2002... and 20009, 30008... etc

A mere 17 of the 81 configurations. Well, let's try another.

13 is a divisor of a mere 41 combinations of numbers with nothing in the middle:

13 26 39 52 65 78 91
104 208 403 507 702 806
1,001 2,002 3,003 4,004 5,005 6,006 7,007 8,008 9,009
20,007 30,004 40,001 60,008 70,005 80,002
100,009 200,005 300,001 500,006 600,002 800,007 900,003
4,000,009 5,000,008 6,000,007 7,000,006 8,000,005 9,000,004

After that, we are back to 10,000,003, 20,000,006, etc because 13 is also a divisor of 999,999.

Well, does this mean that there is no prime number which is a divisor of a full set of possible configurations of numbers with nothing in the middle?

Well, let's try 17:

17 34 51 68 85
102 204 306 408 901
1,003 2,006 3,009 6,001 7,004 8,007
20,009 30,005 40,001 70,006 80,002
200,005 500,004 700,009 800,003
1,000,008 3,000,007 5,000,006 7,000,005 9,000,004
2*107+7 3*107+2 5*107+9 6*107+4 9*107+6
108+1 2*108+2 3*108+3    4*108+4        5*108+5        6*108+6        7*108+7        8*108+8        9*108+9    
2*109+3 4*109+6 6*109+9 7*109+2 9*109+5
4*1010+9 5*1010+7 6*1010+5 7*1010+3 8*1010+1
3*1011+8 4*1011+5 5*1011+2 9*1010+7
1012+4 2*1012+8 5*1012+3 6*1012+7 9*1011+2
1013+6 3*1013+1 4*1013+7 6*1013+2 7*1014+8 9*1012+3
1014+9 2*1014+1 4*1014+2 6*1014+3 8*1014+4
1015+5 4*1015+3 5*1015+8 7*1015+01 8*1015+6
8*1016+9 9*1016+8
And then we are back to 1017+7, and the cycle starts again.

So, indeed, 17 is the first prime number which is a divisor of the full set of possible configurations of numbers with holes in the middle.

There, I thought you needed to know that.

Comments

( 1 comment — Leave a comment )
kevin_standlee
Nov. 20th, 2016 07:20 pm (UTC)
*boggles*
( 1 comment — Leave a comment )

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