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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?

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.

Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Six Times Five

Age 11 to 14
Challenge Level

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

$\frac19$ of all six digit numbers start with a 5. So 100,000 six digit numbers are of the form 5******

This leaves 800,000 numbers that do not start with a 5.

$\frac1{10}$ of the remaining numbers have a 5 in the ten-thousands column, so we need to subtract 80,000 from 800,000 leaving 720,000.

$\frac1{10}$ of the remaining numbers have a 5 in the thousands column, so we need to subtract 72,000 from 720,000, leaving 648,000.

$\frac1{10}$ of the remaining numbers have a 5 in the hundreds column, so we need to subtract 64,800 from 648,000, leaving 583,200.

$\frac1{10}$ of the remaining numbers have a 5 in the tens column, so we need to subtract 58,320 from 583,200 leaving 524,880.

$\frac1{10}$ of the remaining numbers have a 5 in the units column, so we need to subtract 52,488 from 524,880, leaving 472,392.

A slightly quicker method would be to multiply by 0.9 instead of subtracting $\frac1{10}$ in each of the above steps.
 

Here is a different solution, 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 .