Expanding and factorising quadratics

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.

Can you produce convincing arguments that a selection of statements about numbers are true?

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

This is a beautiful result involving a parabola and parallels.

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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?

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.

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

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?