The number 545 has the curious property that — after replacing any single digit by another arbitrary digit (from 0 to 9; it can be a leading 0 or just the same digit) — the result is not divisible by 11.
Is there a positive integer with this property having an even number of digits?
I suspect all numbers with this nondivisibility have the form of ...ABABABA so that is what kind of numbers I will search for.
Call the number N. Let N mod 11 = R.
Changing one of the A digits will change the value of R. The value of R will change directly with the change in A. For nondivisibility by 11 to occur, the smallest value of R must be 1, occuring when the new digit is 0; and the largest value of R must be 10, occuring when the new digit is 9. This then implies that N mod 11 = A+1.
Changing one of the B digits will also change the value of R. The value of R will change inversely with the change in B. For nondivisibility by 11 to occur, the smallest value of R must be 1, occuring when the new digit is 9; and the largest value of R must be 10, occuring when the new digit is 0. This then implies that N mod 11 = 10-B.
Then N mod 11 = A+1 and N mod 11 = 10-B together imply A+1=10-B, or A+B = 9. Then there are 10 cases to try. For each case I list the smallest number I found in that case which has the nondivisibility property:
A,B=0,9: 909090909090
A,B=1,8: 1818181
A,B=2,7: 27272
A,B=3,6: 636363636363
A,B=4,5: 545454545454
A,B=5,4: 545
A,B=6,3: 636363636
A,B=7,2: 272727272727
A,B=8,1: 181818181818
A,B=9,0: 90909090909
This list has five 12-digit numbers with the nondivisibility by 11 property.
Solution
