### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### 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.

# Multiple Surprises

##### Stage: 3 Challenge Level:

Here are some challenges involving consecutive numbers and multiples.

Can you find three consecutive numbers where the first is a multiple of 2, the second is a multiple of 3 and the third is a multiple of 4?

Can you find several examples?
What do you notice?

What if the first is a multiple of 3, the second is a multiple of 4 and the third is a multiple of 5?

What if the first is a multiple of 4, the second is a multiple of 5, and the third is a multiple of 6?

Is there a way to find sets of four consecutive numbers which are multiples of 2, 3, 4 and 5 (in this order)?

Or five consecutive numbers which are multiples of 2, 3, 4, 5 and 6 (in this order)?

With thanks to Don Steward, whose ideas formed the basis of this problem.