Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Can you prove our inequality holds for all values of x and y between 0 and 1?

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Can you find the hidden factors which multiply together to produce each quadratic expression?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .

This is a beautiful result involving a parabola and parallels.

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

How do scores on dice and factors of polynomials relate to each other?

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

If you plot these graphs they may look the same, but are they?

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.