### 14 Divisors

What is the smallest number with exactly 14 divisors?

### 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.

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?