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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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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?

More Mathematical Mysteries

Age 11 to 14 Challenge Level:

Meg offers the following solution:

If you take any 3 digit number, and then rearrange the digits of the number and find the difference between the two numbers, the number will always be a multiple of 9.

Think about the digits as being a,b and c.
These can be rearranged and subtracted in the following ways:
abc-acb= (10b+c)-(10c+b)=9(b-c)
abc-bca=(100a+10b+c)-(100b+10c+a)= 99a-90b-9c=9(11a-10b-c)

The digits of any multiple of 9 can always be reduced to give 9. eg
18: 1+8= 9
27: 2+7 = 9
99:9+9 = 18; 1+8 = 9

So whichever numbers you use, you will always end up with 9.
4 translates to D using the code
The only countries beginning with D are Denmark, Djibati, Dominica and the Dominican Republic. Most people would choose Denmark.
You'd be hard-pushed to think of another jungle animal beginning with E other than elephant.
Animals beginning with K are not many in number, so Koala may come to mind.
This leads to a fruit beginning with A, which is most obviously apple.