What six digit number added to itself five times will give a sum each time having the same 6 digits as the original number?

(It's not 000000.)

(In reply to Wild guess by Penny)

Answer same as Penny's:

We have a 6 digit number that after being multplied with 6, still is a 6 digit number.  This means that the number has to start with 1, any other number would become a 7 digit number.

If there is a 1 in the six digit number, there also needs to be at least one (odd) digit to get the number when 1 is at the end of the six digit number. The odd digit can't be a 3, as this would give: 3, 6, 9, 2, 5 as last digit.  Same goes for 5 and 9, which means that 7 is the last digit and the other four digits are 7*2-10=4 ; 7*4-20=8: 7*5-30=5; 7*6-40=2
Or the number is 1????7, where ???? are 2,4,5 and 8

The first two ?? can't be 5 or 8, otherwise the first digit of the second 6 digit number would be a 3.  This leaves us with either 142 or 124.  If the first two digits where12, then the second 6 digit number would start with 24, which is not possible, as we need the 4 at the end as last digit (2*7).

So we now have 142??7.
The place of the 8 and 5 can be found easily when looking at the 4th 6digit number: 4 times 587 would need a 3, so it has to be 1428757

Posted by Hugo on 2005-07-04 19:41:48