Powers and roots

problem

Power Crazy

Age

11 to 14

Challenge level

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
problem

Archimedes and Numerical Roots

Age

14 to 16

Challenge level

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
problem

Rachel's Problem

Age

14 to 16

Challenge level

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!
problem

Deep Roots

Age

14 to 16

Challenge level

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$
problem

Lastly - Well

Age

11 to 14

Challenge level

What are the last two digits of 2^(2^2003)?
problem

Bina-Ring

Age

16 to 18

Challenge level

Investigate powers of numbers of the form (1 + sqrt 2).
problem

St Ives

Age

7 to 14

Challenge level

How many were going to St Ives?
problem

Largest Number

Age

11 to 14

Challenge level

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.
problem

Maundy Money

Age

11 to 14

Challenge level

How much money did the Queen give away in pence as a power of 2?
problem

Age of Augustus

Age

11 to 14

Challenge level

The English mathematician Augustus de Morgan has given his age in algebraic terms. Can you work out when he was born?