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

Why do this problem?

This problem is short and encourages students to think about the meaning of place value and engages their logical thinking. It could be used as a starter to engage pupils as they come in to the classroom, though there are good extensions available for a full lesson.

Possible approach

Put the problem on the board to allow pupils to familiarise themselves with the problem.
Discuss as a group the possible forms of proper 6 digits numbers. Students might like to decide whether numbers starting with zero count as a proper six-digit number (no!).
Allow students to experiment to try to determine the number of possible answers.

A good problem solving strategy is to make the problem smaller, e.g.how many three- (or four-) digit numbers do not contain a $5$, then to work out how to extend the solution method.

Key questions

• Is $000001$ a six-digit number?
• How many six-digit numbers are there?
• How many choices do we have for the first digit?
• How many choices do we have for the second digit?

Possible extension

• How many six-digit numbers do not contain a $5$ or a $7$?
• How many six-digit numbers are there for which the digits increase from left to right (such as $134689$ or $356789$)?
• How many numbers less than $10$ million do not contain a $5$?
• Will your methods extend to similar problems? if so, can you express them algebraically?
• What other [interesting] questions could you ask starting "How many six-digit numbers..."?

Possible support

• You could ask the almost equivalent question "How many six-figure telephone numbers do not contain a $5$?". This encourages student to imagine dialling a number in sequence, which will may help them to see the different choices which can be made at each step of the process.
• Encourage students to adapt the problem to make it accessible: fewer digits, how many six-digit numbers are a multiple of $10$ (probably seen as a number with $0$ as last digit) or even, or a mult of $5$, or square, etc.