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.

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:

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 .