is divisible by 7, prove that it is also divisible by 11, 13, and 15,873. In the

is divisible by 7, prove that it is also divisible by 11, 13, and 15,873.

In the sequence 1, 11, 111, 1111, …:

Prove there exist two terms whose difference is divisible by 196,673.

Deduce that there exists a repunit divisible by 196,673.

For any natural number a not divisible by 2 or 5, prove there exists a natural b such that the product ab is a repunit (written using only the digit 1).

If natural numbers a and m are coprime, prove there exists n such that an − 1 is divisible by m.

C. Cyclic digit moves and special base‑10 constructions

A k-digit multiple of 13 has its first digit moved to the end. For which k is the resulting number still a multiple of 13?

Example: 503,906 → 39,065 remains divisible by 13; 7,969 → 9,697 does not.

Find a six‑digit decimal number that becomes 5 times smaller when its first digit is moved to the end of the number.

The last digit of a (decimal) number is