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Resources tagged with Properties of numbers similar to An Introduction to Proof by Contradiction:

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

An Introduction to Irrational Numbers

Stage: 4 and 5

Tim Rowland introduces irrational numbers

What Are Numbers?

Stage: 2, 3, 4 and 5

Ranging from kindergarten mathematics to the fringe of research this informal article paints the big picture of number in a non technical way suitable for primary teachers and older students.

Stage: 4 Challenge Level:

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

Babylon Numbers

Stage: 3, 4 and 5 Challenge Level:

Can you make a hypothesis to explain these ancient numbers?

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?

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?

Fracmax

Stage: 4 Challenge Level:

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

Summing Consecutive Numbers

Stage: 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Cinema Problem

Stage: 3 and 4 Challenge Level:

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly Â£100 if the prices are Â£10 for adults, 50p for pensioners and 10p for children.

Lastly - Well

Stage: 3 Challenge Level:

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

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?

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?

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.

Six Times Five

Stage: 3 Challenge Level:

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

Sept 03

Stage: 3 Challenge Level:

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

Magic Letters

Stage: 3 Challenge Level:

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Oh! Hidden Inside?

Stage: 3 Challenge Level:

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

Repetitiously

Stage: 3 Challenge Level:

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

Cogs

Stage: 3 Challenge Level:

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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.

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?

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?

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?

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?

Filling the Gaps

Stage: 4 Challenge Level:

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

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?

Pair Products

Stage: 4 Challenge Level:

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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?

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?

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?

Odd Differences

Stage: 4 Challenge Level:

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

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?

Rachel's Problem

Stage: 4 Challenge Level:

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

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.

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.

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?

Mini-max

Stage: 3 Challenge Level:

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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. . . .

Whole Numbers Only

Stage: 3 Challenge Level:

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

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?

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?

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.

Triangular Triples

Stage: 3 Challenge Level:

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

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...

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.

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" .

Enriching Experience

Stage: 4 Challenge Level:

Find the five distinct digits N, R, I, C and H in the following nomogram

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?