Choose two digits and arrange them to make two double-digit
numbers. 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?
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.
What is the sum of all the digits in all the integers from one to
Write down any nine digit number which uses each of the digits
1, 2, 3, ..., 9 once only.
Change the number by re-writing it with the very first digit as the
units digit at the end and otherwise keeping the digits in the same
For example 354218697 becomes 542186973.
This is called a cyclic permutation of the digits. By now you will
have two numbers written down.
Repeat the cyclic permutation again and again writing down all the
new numbers you obtain until you get back to your first number. Add
up these nine numbers.
Prove that, whatever number you chose originally, the total
obtained in this way is the same.
[Note: The digits can be cyclically permuted in the opposite
direction and, more generally, abcdefghi and bcdefghia are cyclic
permutations of each other].