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?
First, why does 545 have this property? (5+5)-(4)=6. Which needs to become either 0 or 11. Changing a 5 can increase this by up to 4 or decrease by 5. Changing the 4 can only do the same. So the 6 (mod 11) is the important difference if we are going to have a number composed of alternating 5's and 4's.
545454545454 has the property and has 12 digits as does its reverse.
5454545454545454545454545 has 17.
You can also append or 454545454 to these.
Is there a number not composed of alternate 4's and 5's? Yes, but I overlooked them. See Brian Smith's Solution.
Edited on August 6, 2017, 2:41 pm
Posted by Jer
on 2017-08-06 09:24:22