Number theory

problem
Novemberish

Age

14 to 16

Challenge level

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.
problem
Mod 7

Age

16 to 18

Challenge level

Find the remainder when 3^{2001} is divided by 7.
problem
There's a limit

Age

14 to 18

Challenge level

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
problem
Diophantine n-tuples

Age

14 to 16

Challenge level

Can you explain why a sequence of operations always gives you perfect squares?
problem
Pythagorean golden means

Age

16 to 18

Challenge level

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
problem
Euler's squares

Age

14 to 16

Challenge level

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
article
An introduction to modular arithmetic

An introduction to the notation and uses of modular arithmetic
$Got \It Article$

article
Got it article

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
article
Powerful properties

Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.
article
About Pythagorean golden means

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?