Always a Multiple?

Think of a two-digit number.
Reverse the digits and add your answer to your original number.

For example, Alison chose 42 and added 24 to get 66.

Try a few examples.
Can you explain why your answer is always a multiple of 11?

Alison and Charlie came up with their own explanations:

If you can't play the videos, you can read a description below.

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 $11(a+b)$.

Here are some similar number tricks.
Can you use Charlie's or Alison's representation to explain how they work?

Take any two-digit number. Reverse the digits, and subtract your answer from your original number. What do you notice?
Take any two-digit number. Add its digits, and subtract your answer from your original number. What do you notice?
Take any three-digit number. Reverse the digits, and subtract your answer from your original number. What do you notice?
Take any five-digit number. Reverse the digits, and subtract your answer from your original number. What do you notice?

Can you make up some number tricks of your own?

Comments

THIS QUESTION

This Question

Really cool, easy to

COOL

it is always a multiple of 11

maths