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.