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Choose two digits and arrange them to make two double-digit numbers, for example:
If you choose $1$ and $2$,
you can make $12$ and $21$
Now add your double-digit numbers.
Now add your single-digit numbers.
Divide your double-digit answer by your single-digit answer.
Try lots of examples. What happens? Can you explain it?
What happens if you choose zero as one of the digits?
Try to explain why.
How does it work if you choose the same digits, for example $3$ and $3$?
What happens if you use negative numbers?
Now choose three digits and arrange them to make six different triple-digit numbers.
Repeat the steps above: add the triple-digit numbers, add the single digits then divide the triple-digit answer by the single-digit answer.
Do you get the same results?
If you're feeling very organised, try more digits and see what happens.
This problem has been adapted from the book "Numbers in Your Head" by John Spooner, published by BEAM Education. This book is out of print but can still be found on Amazon.
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.
Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?