### 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.