Divisibility

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Use properties of numbers to work out whether you can satisfy all these statements at the same time.

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?