### 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.

### Skeleton

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

# More Magic Potting Sheds

##### Age 11 to 14 Challenge Level:

This problem follows on from Magic Potting Sheds

After a year of successful gardening using his magic doubling shed (introduced in Magic Potting Sheds), Mr McGregor buys a new shed that trebles the number of plants in it each night. Use the interactivity to investigate how many plants he needs this time to get the same number in each garden. What is the smallest number of plants he could use?

Move the slider at the bottom to choose the Magic Multiplier for the shed. Type in the input box or use the slider below it to choose how many plants to buy.

Can you predict how many plants he would need on the first day and how many he should plant each day if he bought a new shed that quadruples the number of plants in it each night?

Use the interactivity to test your prediction.

Mr McGregor is so successful that he decides to plant more gardens. He can still only plant one garden each day.

Use the interactivity to change the number of gardens and investigate how many plants he should use for each of the different potting sheds.

What do you find?
Can you find a general rule?
Can you explain why your rule works?

Unfortunately, Mr McGregor suffers an attack from evil magic slugs that eat half of the plants in his (non-magic) potting shed each night. He still wishes to plant the same number of plants in each garden.

How many plants does he need on the first day this time, and how many should he plant each day? (Remember that he can only plant whole numbers of plants!)