What Is The Smallest Nine Digit Number That Is Divisible By 11? No Digit Should Be Repeated

[oflakes]

What is the smallest nine digit number that is divisible by 11? No digit should be repeated!

[Prudie]

I think you'll have to be satisfied you are correct FF without confirmation, this poster looks like a one hit wonder.

[fiction-factory]

I'm interested to see if anyone can come up with an answer and a quicker method

[jim360]

I thought only about exhaustively trying all nine-digit numbers starting at 987564321. It may not be the most efficient but at least it kept me too busy to post anything today!

[JimF]

"102347586"
On further investigation, I think F-F is correct.

[JimF]

"a quicker method"

No idea if it's quicker or not, but I just used Excel.

[fiction-factory]

I started with 102345678 and the first multiple of 11 I came to was 102345683, so I used a spreadsheet to count up in 11s from there and then started eliminating sets of last few digits

[JimF]

I did a similar thing:

A1: 10245678
A2: =A1/11
A3: =SUM(IF((LEN(A1)-LEN(SUBSTITUTE(A1,{1,2,3,4,5,6,7,8,9,0},""))),1,0))
A2: =A1+1
B2: Copy A1
B3: Copy A3

Copy row 2 for as long as it takes to find the first record where column B is an integer and column C has a value of 9.

[JimF]

B2: Copy A2

[NickorWan]

You can prove that 123475869 is divisible by 11 by checking the alternate sum is divisible by 11, so 1-2+3-4+7-5+8-6+9. I wonder could that be worked in reverse in some way.

[fiction-factory]

Yes, that's the sort of divisibility test I was looking for. I know the obvious ones like testing for divisibility by 3 or 9 but I wasn't sure what the rule would be for 11 so that's interesting, Nick

[Prudie]

That's interesting Nick but when does that start working as it doesn't for the double digits, nor 3 except 209, that's where I stopped checking...

[NickorWan]

I can't actually remember, Prudie, to be hones, I'll have to check myself. I only remember it as a guide for big numbers. I might even be confusing it with something else, that does happen these days, but it seemed to work with the ones I tried.

[NickorWan]

It might even be a case of that method proves the number is divisible by 11, but doesn't say that a number that doesn't do that isn't, which would be naff all use in this case!

I have been doing maths refreshers lately thanks to Numberphile, but I have a long way to go yet:)

[aelmpvw]

my contribution is that the answer will begin with 1

[ToraToraTora]

ok doing what programmers do when their maths skills run out I wrote a simple program to do it by brute force, the answer is 123475869, assuming no zero is allowed. For those that can stand the excitement here is my code, It's in an IBM mainframe language called REXX.
https://ibb.co/m9mr1xG

[ToraToraTora]

so yes FF was correct earlier though no idea how he got there.

[fiction-factory]

Thanks TTT. For the 'no zero' solution I started with 123456789 then counted on until I found the first number after that divisible by 11, which was 123456795.
I then used Excel to count on in 11s to list the next few thousand multiples of 11.
I then worked my way down and deleted big blocks by eliminating those where the last 3 digits contained anything other than a combination of 789.
Then when I'd eliminated all those I looked at those starting 12345 and deleted all those with anything other than combinations of 6, 7, 8 and 9.
Then I looked at those starting with 12346 and eliminated everything by inspection as there were none ending in combinations of 5,7,8,9.

So then I did those starting with 12347 and looked for the first item ending in a combination of 5,6,8,9. That brought me to 123475869

So a combination of narrowing it down logically plus brute force and inspection.

[ToraToraTora]

well FF you have more patience than me! I just wrote the above program, which you can actually do in VB if you have that option in excel

[fiction-factory]

Yes, I tend to use only the more basic Excel functions now. It took me maybe 20-30 minutes to eliminate the nonqualifying numbers by inspection and then go back and double check I hadn't missed a solution, but it probably would have taken me a lot longer to remember how to write the necessary program.
Many many years ago I used to do these things by writing a programme in Basic on my Commodore 64 but I used to have to leave it running overnight

[ToraToraTora]

this problem would not have taken overnight, it ran in 0.04 of a CPU second.