Number theory

problem

Mod 7

Age

16 to 18

Challenge level

Find the remainder when 3^{2001} is divided by 7.
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

Strange Numbers

Age

11 to 14

Challenge level

All strange numbers are prime. Every one digit prime number is strange and a number of two or more digits is strange if and only if so are the two numbers obtained from it by omitting either its first or its last digit. Find all strange numbers.
problem

A One in Seven Chance

Age

11 to 14

Challenge level

What is the remainder when 2^{164}is divided by 7?
problem

The Public Key

Age

16 to 18

Challenge level

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.
problem

Marbles

Age

11 to 14

Challenge level

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
problem

More Marbles

Age

11 to 14

Challenge level

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
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...
problem

Never Prime

Age

14 to 16

Challenge level

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.
problem

Ordered Sums

Age

14 to 16

Challenge level

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.