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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?

Six Times Five

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

No correct solution to this problem was originally received. However, Mary of Birchwood Community High School gave a sound argument that just needs some adaptation. Thank you Mary..

Firstly Mary considered the number of six digit numbers - this is 900,000.

10% of all six digit numbers start with a 5. So 90,000 six digit numbers are of the form 5******

This leaves 810,000 numbers that do not start with a 5. How many of these have a 5 as the second digit??

And so on.....

Here is a solution to this toughnut from Junwei of BHASVIC

Let the six digits number is abcdef, which a, b, c, d ,e, f represent a digit respectively.

For a, neither 0 nor 5 could place in it, thus, 8 digits are available here (1,2,3,4,6,7,8,9)

For b, c, d, e and f, they can't contain 5, hence, 9 digits are available for them (0,1,2,3,4,6,7,8,9)

Therefore, the no. of six digits number which does not contain any 5 is

8 * 9 * 9 * 9 * 9 *9 =472392 .