Prove that the numbers that begin with 1 and end with 1, with any number of 2s in the middle, and all 1s and 2s separated by 00 are composite.
For example:
1002001
1002002001,
1002002002001,
1002002002002001
Each number is divisible by 11.
To see why, recall the divisibility test for 11: the sum of the even power decimal digits minus the sum of the odd power digits must itself be divisible by 11.
Example: 3828.
Even digits - odd digits = 16 - 5 = 11.
Hence 3828 is divisible by 11.
Applying this test to the numbers above, notice that, since the double zeroes occupy an odd and an even position, they can be removed without affecting divisibility by 11. So we obtain:
121,
1221,
12221,
122221.
Now it is a simple matter to confirm that the first two numbers are divisible by 11. It follows that the other numbers in the series are divisible by 11, since they can be constructed by inserting a number of double twos (one even and one odd digit, again) into one of 121 or 1221.
You can also show that each number is divisible by 1001 by setting x = 1000 in the identity:
(1 + x)(1 + x + ... + x^n) = 1 + 2x + ... + 2x^n + x^(n+1).
Divisibility
