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Broad Topics > Numbers and the Number System > Properties of numbers

Helen's Conjecture

Stage: 3 Challenge Level:

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Thirty Six Exactly

Stage: 3 Challenge Level:

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

Two Much

Stage: 3 Challenge Level:

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

One to Eight

Stage: 3 Challenge Level:

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

X Marks the Spot

Stage: 3 Challenge Level:

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

Oh! Hidden Inside?

Stage: 3 Challenge Level:

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Stage: 3 Challenge Level:

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

Slippy Numbers

Stage: 3 Challenge Level:

The number 10112359550561797752808988764044943820224719 is called a 'slippy number' because, when the last digit 9 is moved to the front, the new number produced is the slippy number multiplied by 9.

Factorial

Stage: 4 Challenge Level:

How many zeros are there at the end of the number which is the product of first hundred positive integers?

Happy Octopus

Stage: 3 Challenge Level:

This investigation is about happy numbers in the World of the Octopus where all numbers are written in base 8 ... Find all the fixed points and cycles for the happy number sequences in base 8.

Four Coloured Lights

Stage: 3 Challenge Level:

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

Like Powers

Stage: 3 Challenge Level:

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n$ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

Times Right

Stage: 3 and 4 Challenge Level:

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

The Patent Solution

Stage: 3 Challenge Level:

A combination mechanism for a safe comprises thirty-two tumblers numbered from one to thirty-two in such a way that the numbers in each wheel total 132... Could you open the safe?

Arrange the Digits

Stage: 3 Challenge Level:

Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

Chameleons

Stage: 3 Challenge Level:

Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .

Six Times Five

Stage: 3 Challenge Level:

How many six digit numbers are there which DO NOT contain a 5?

A Little Light Thinking

Stage: 4 Challenge Level:

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Not a Polite Question

Stage: 3 Challenge Level:

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

Sept 03

Stage: 3 Challenge Level:

What is the last digit of the number 1 / 5^903 ?

Lastly - Well

Stage: 3 Challenge Level:

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

Filling the Gaps

Stage: 4 Challenge Level:

Which numbers can we write as a sum of square numbers?

Alphabet Soup

Stage: 3 Challenge Level:

This challenge is to make up YOUR OWN alphanumeric. Each letter represents a digit and where the same letter appears more than once it must represent the same digit each time.

Lesser Digits

Stage: 3 Challenge Level:

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

Whole Numbers Only

Stage: 3 Challenge Level:

Can you work out how many of each kind of pencil this student bought?

Counting Factors

Stage: 3 Challenge Level:

Is there an efficient way to work out how many factors a large number has?

Water Lilies

Stage: 3 Challenge Level:

There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?

Triangular Triples

Stage: 3 Challenge Level:

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

A Long Time at the Till

Stage: 4 and 5 Challenge Level:

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Even So

Stage: 3 Challenge Level:

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

One or Both

Stage: 3 Challenge Level:

Problem one was solved by 70% of the pupils. Problem 2 was solved by 60% of them. Every pupil solved at least one of the problems. Nine pupils solved both problems. How many pupils took the exam?

N Is a Number

Stage: 3 Challenge Level:

N people visit their friends staying N kilometres along the coast. Some walk along the cliff path at N km an hour, the rest go by car. How long is the road?

Elevenses

Stage: 3 Challenge Level:

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Small Change

Stage: 3 Challenge Level:

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

The Codabar Check

Stage: 3

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

Rachel's Problem

Stage: 4 Challenge Level:

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

Clever Carl

Stage: 2 and 3

What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.

Guess the Dominoes

Stage: 1, 2 and 3 Challenge Level:

This task depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together.

Unit Fractions

Stage: 3 Challenge Level:

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

Stage: 3 Challenge Level:

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Writ Large

Stage: 3 Challenge Level:

Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

Got it Article

Stage: 2 and 3

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

Really Mr. Bond

Stage: 4 Challenge Level:

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

See the Light

Stage: 2 and 3 Challenge Level:

Work out how to light up the single light. What's the rule?

Difference Dynamics

Stage: 4 and 5 Challenge Level:

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

An Introduction to Irrational Numbers

Stage: 4 and 5

Tim Rowland introduces irrational numbers

Generating Triples

Stage: 4 Challenge Level:

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Babylon Numbers

Stage: 3, 4 and 5 Challenge Level:

Can you make a hypothesis to explain these ancient numbers?