Q:

Show that a given number has two identical digits(Kosovo TST 2011)

Starting with the number $7^{1996}$ we remove its first digit, and then add that digit to the rest of the number. This process continues until the result has ten digits. Show that the resulting number has two of its digits the same.

A:

Hint: if the leftover number has all of its digits different, what is its digit-sum?  What does that tell you about its divisibility?  Can you show that the process specified doesn't change the relevant divisibility property?

