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Double Digit

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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.

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Digit Sum

What is the sum of all the digits in all the integers from one to one million?

Cycle It

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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 order.

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].