This is a math activity that needs you to use your best investigation skills. So get out your paper and pencil and put on your thinking cap. (And you may wish to consider a calculator, but it won’t be of much use some times.)
Let a rep-digit number be any number that consists of a string of only one kind of digit, but which may be repeated as much as desired. Hence, the name rep-digit is short for repeated-digit.
For example, 111, 44444, 66, 22222222, and 8 are all rep-digit numbers. Their lengths (or if you prefer, orders) are 3, 5, 2, 8, and 1, respectively.
Show that any rep-digit number made up of ones (i.e. 1, 11, 111, etc.) when divided by 7 leaves a zero remainder when the length of the number is a multiple of six; otherwise such divisions will leave any remainder (less than 7) except 3.
What corresponding peculiarities can you discover for the remainders when you divide other rep-digit numbers by 7. That is, sets of numbers of the form d, dd, ddd, … , where d can be any whole number from 2 to 9.
Now combine the ideas of #1 and #2 and use other numbers besides 7 as the divisors. What patterns can you find? What overall observations can you make? Pay special attention to divisors that are primes.
I especially recommend the primes 13, 17, 19, and 23. Make a string of 1’s long enough until you have no remainder at all with those numbers as the divisors. If you begin by putting the “unnecessary” 0’s in the quotient position, it will help in discovering some unique facts of the “split-half” variety. (See the number 142857 in Special Numbers.)
0 0 8 5 13) 1 1 1 1 ... 1 0 4 7 1 6 5 6
One last piece of advice: if you’re doing this long division work “by hand”, use a piece of ordinary graph paper, and put one digit in each square as you move through the procedure. It helped me, so I think it would help you. 😉