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Double Digit

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?

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Repeaters

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?

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Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2
$3$

The units digit of the product is $3$; therefore the remainder when that number is divided by $5$ is also $3$.


This problem is taken from the UKMT Mathematical Challenges.