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Can you explain the strategy for winning this game with any target? ### Counting Factors

Is there an efficient way to work out how many factors a large number has? ### 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.

### Why do this problem?

This problem requires plenty of accurate adding! Although the ability to do division is called for, calculators could be used to perform the operation as well as to check results.

The investigation leads learners to generate for themselves the rule for divisibility by $9$ - that if the digits in a number add to $9$ or a multiple of $9$.

### Key questions

Is this number a multiple of $9$?
How many $2$-digit numbers have you found that are divisible by $9$?
What happens if you just use the numbers from $1$ to $8$?
More able learners could explore what multiples of $9$ they can and cannot make using all the digits $1$ to $9$ once and once only. These will be between $45$ (the result of adding all nine digits as $1$-figure numbers) and $987654321 + 1$. Repeat with he set of numbers $1$ to $8$.
Suggest finding different $2$-digit numbers the set of digits $1$ to $9$, and then total these adding in the 'extra' digit and work from this total.