### 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.

### Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

### Like Powers

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n$ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

# Multiples Sudoku

### Why do this problem?

This Sudoku offers an engaging context which requires students to think logically and apply their knowledge of factors and multiples.

### Possible approach

These printable resource may be useful: Multiples Sudoku
Multiples Sudoku Journey

If your students do not know the rules of Sudoku then set aside a little time for them to become familiar with the 'standard' Sudoku.

Work together with the class filling in a few cells to make sure everyone understands the rules - the 3rd, 5th and 7th rows offer opportunities for filling cells easily at the early stages.

Then hand out the sheets and invite students to work in pairs, emphasising that they must convince each other that their suggestions are correct, before anything gets added onto their papers.

### Key questions

Some clues have lots of possibilities and some have few. Which are which?
Which are the most helpful clues to begin?

### Possible support

Provide students with this possible journey through Multiples Sudoku and suggest they try to retrace the route.

For other Factors and Multiples problems that might help to prepare your students for this task, see Missing MultipliersDozens, the Factors and Multiples Game, and the Factors, Multiples and Primes Short Problems collection.

### Possible extension

Students may wish to work on the similar but more challenging tasks Diagonal Product Sudoku and Product Sudoku.

For more challenges on Factors and Multiples, see Gabriel's Problem or the Factors, Multiples and Primes Short Problems collection.