This problem is in two parts. The first part provides some building blocks which will help you to solve the final challenge. These can be attempted in any order. Of course, you are welcome to go straight to the Final Challenge!
Click a question from below to get started.

Question A
Choose any two numbers from the $7$ times table. Add them together. Repeat with some other examples. Notice anything interesting?

Now do the same with a different times table. What do you notice this time? Convince yourself it always happens.

Question B
Choose two digits and arrange them to make two double-digit numbers.

For example, if you choose $5$ and $2$, you can make $52$ and $25$.

Now add your two-digit numbers.

Repeat with some other examples.

Notice anything interesting? Convince yourself it always happens.

Question C
Look at this sequence of numbers: $11, 101, 1001, 10001, 100001, ...$

Divide numbers in this sequence by $11$, WITHOUT using a calculator.

Notice anything interesting? Convince yourself it always happens.

FINAL CHALLENGE
Take any four-digit number, move the first digit to the 'back of the queue' and move the rest along. For example $5238$ would become $2385$.

Now add your two numbers.

Is the answer always a multiple of $11$? Can you convince yourself?

What happens when you do this with three-digit numbers? Five-digit numbers? Six-digit numbers? 38-digit numbers ... ?

Prove your findings!

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