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Take any four-digit number.
Create a second number by moving the first digit to the 'back of the queue' and moving the rest along. 
Now add your two numbers.
I predict your answer will be a multiple of 11...

animation showing 5238 + 2385 = 7623 

Try it a few times. Is the answer always a multiple of $11$?
Can you explain why?

Click below to see what Samira noticed:

"I started with 5 thousands, 2 hundreds, 3 tens and 8 units.
After I moved the digits along, my new number had 2 thousands, 3 hundreds, 8 tens and 5 units. 

I wonder if this can help me explain what's happening?"

Click below to see what Jay noticed:

"I picked 1000 as my first number, so my second number was 0001 and the total was 1001.
I know 1001 is a multiple of 11 because it is 1100-99, and 1100 and 99 are both multiples of 11."

Do these observations help you to explain what's going on?

What if you start with a three-digit number?
Or a five-digit number?
Or a six-digit number?
Or a 38-digit numbers ... ?

Can you prove your findings?

You may be interested in this article on Divisibility Tests.

Click here for a poster of this problem.