### 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.

### Oh! Hidden Inside?

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

# American Billions

##### Stage: 3 Challenge Level:

Alison and Charlie are playing a divisibility game with a set of $0-9$ digit cards.

They take it in turns to choose and place a card to the right of the cards that are already there.

• After two cards have been placed, the two-digit number must be divisible by $2$.
• After three cards have been placed, the three-digit number must be divisible by $3$.
• After four cards have been placed, the four-digit number must be divisible by $4$.

And so on!

They keep taking it in turns until one of them gets stuck.

Alison places the $5$.
Charlie puts down the $8$ to make $58$, which is a multiple of $2$.
Alison puts down the $2$ to make $582$, which is a multiple of $3$.
Charlie puts down the $0$ to make $5820$, which is a multiple of $4$.
Alison now has to choose from $1, 3, 4, 6, 7,$ or $9$ to make a multiple of $5$.
Convince yourself that Alison is stuck, and that Charlie has won.

Play the game a few times on your own or with a friend.

After a while, Charlie and Alison decide to work together to make the longest number that they possibly can that satisfies the rules of the game.

They very quickly come up with the five-digit number $12365$. Can they make their number any longer using the remaining digits? When will they get stuck?

What's the longest number you can make that satisfies the rules of the game?

Is it possible to use all ten digits to create a ten-digit number?
Is there more than one solution?

Please send us your explanation of the strategies you use to create long numbers.