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'Always a Multiple?' printed from http://nrich.maths.org/
Charlie said: "Alison, think of a two-digit number. Reverse the digits and add your answer to your original number. I bet your answer is a multiple of 11."
Alison chose 42, added 24 and got the answer 66: "It is! How on earth did you know that?"
Charlie said: "I'm not sure. Let's try to work it out."
Alison arranged multilink to show four tens and two units for 42, and two tens and four units for 24.
She then put the four units with the four tens, and the two units with the two tens, giving six lots of eleven.
Charlie imagined a two-digit number $ab$, where $a$ represents the number in the tens column, and $b$ respresents the number in the units. This can be written as $10a+b$. Similarly, $ba$ can be written as $10b+a$.
Charlie added these together to get $11a+11b$, which he wrote as $11(a+b)$.