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!

[jim360]

Prudie: Wan's 11 "alternating digits sum" trick works for all numbers including double digits, as 0 is divisible by 11.

[fiction-factory]

Yes, I looked Nic's method up Jim and maybe I'd forgotten about it but seems obvious now- based on modular arithmetic.

[Prudie]

I dismissed that as we have decided, to aid calculation, that 0 divided by anything is exactly divisible and = 0 but it's not real :-)
If it were then there is an argument for:
0/1 =0/2=0/3=0/78=0/1023 etc hence 1=2=3=78=1023...since if a/b = a/c then b=c (I know, unless a is zero, but 'we' have defined that)

[jim360]

Except that misses the point. Compare 22 -> 2-2 = 0 with 23 -> 2-3 = -1, or similarly 32 with 33 (alternating sum of 1 vs. zero). Similarly 132 -> 1 - 3 + 2 = 0, and indeed 132 divides exactly by 11, whereas 123 -> 1 - 2 + 3 = 2 and 123 doesn't divide exactly by 11.

[NickorWan]

Going back to my wondering then, would it have helped answer the OP? Given the lowest number possible from the digits is 1-2+3-4+5-6+7-8+9 = 5, would you have found it easy to work out the changes needed so that it added up to a multiple of 11 leaving the smallest number?

[Peter Pedant]

I think in fairness FF
as a maff freek - or maths master as we used to call them
you should tell us how you did it

zebo SAYS that numberacy ability to add, is different to maff
I thought of you because you have pretty impressively Done This Before - so how dja do it?

[Peter Pedant]

o god I thought you had done it by alternate digits

[jim360]

List of tests for divisibility numbers that I can be bothered to work out. All numbers in base ten are written ....dcba (eg 1345 has d = 1, c = 3 etc):

1: automatic
2: a
3: sum of digits
4: 2b + a
5: a = 0 or 5
6: passes test for 2 and 3
7: a + 3b + 2c - d - 3e - 2f + g... or (abc) - (def) + (ghi) - ...
8: 4c + 2b + a
9: sum of digits
10: a=0
11: alternating sum of digits
12: passes test for 3 and 4
13: (abc) - (def) + (ghi) - ...

[Peter Pedant]

christ
perhaps I wdnt have bloomed on the maff course I got onto 2012
( cancer stopped play (*))

(*) really bad blood cancer - not just those lumps people get

[jim360]

Typo in my last. Should be (cba) - (fed) + (ihg) etc for 7 and 13.

[oflakes]

OK THE CORRECT ANSWER IS: 123475968
This is the smallest possible nine digit number which is divisible by 11 with no repeated digits. And i forgot to mention zero is not one of the digits.

[oflakes]

THE METHOD:
We should use digits 1-9. And to be divisible by 11, the sum of alternate digits - other digits should be eleven.
1 2 3 4 7 5 9 6 8
1+3+7+9+8=28
2+4+5+6=17
28-17=11
You have to just find the combination in which the final number u get is the smallest. As u can see this the correct combination.

[fiction-factory]

Hi. You say the answer is 123475968
What was wrong with my suggestion of 123475869 which is smaller than yours?

[oflakes]

yeah ff my bad.
THE ANSWER IS 123475869
FF GOT IT FIRST. :)