A special congratulations to Yat Sum from from Cambourne Village College for being the only person to successfully solve part i, with a partial solution on ii.
Solution for i
For any -digit positive integer
with digits
, we can express
as:
Rewrite



Each term




Solution for ii
Lemma 1: Any number in the form with an even number of digits is divisible by 11.
Performing long division of by 11 shows that 11 divides each block of two zeros, leaving a remainder of 1 carried over at each step.
With an even number of digits, this remainder of 1 continues to the last digit and balances out exactly, showing that all even-length numbers of the form are divisible by 11.
Lemma 2: Any even-length number in the form is divisible by 11.
Explanation:
Each block of two nines (like 99) is divisible by 11. So any even-length sequence of 9’s, like , is divisible by 11.
Proof:
Let . Rewrite
by decomposing it as a sum of terms in the form
and
:
The terms in the form



is divisible by 11.