What is the smallest number with exactly 14 divisors?
This package contains a collection of problems from the NRICH
website that could be suitable for students who have a good
understanding of Factors and Multiples and who feel ready to take
on some. . . .
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
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?
Bernard Bagnall recommends some primary school problems which use
numbers from the environment around us, from clocks to house
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?
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.
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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
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.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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" .
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. . . .
Nearly all of us have made table patterns on hundred squares, that
is 10 by 10 grids. This problem looks at the patterns on
differently sized square grids.
Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
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.
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
A case is found with a combination lock. There is one clue about the number needed to open the case. Can you find the number and open the case?
I am thinking of three sets of numbers less than 101. Can you find
all the numbers in each set from these clues?
Can you find any perfect numbers? Read this article to find out more...
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.
I am thinking of three sets of numbers less than 101. They are the
red set, the green set and the blue set. Can you find all the
numbers in the sets from these clues?
This is a game in which your counters move in a spiral round the snail's shell. It is about understanding tens and units.
Guess the Dominoes for child and adult.
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?
This task depends on learners sharing reasoning, listening to
opinions, reflecting and pulling ideas together.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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?
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?
Find out about palindromic numbers by reading this article.
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?
A cinema has 100 seats. Show how it is possible to sell exactly 100
tickets and take exactly £100 if the prices are £10 for
adults, 50p for pensioners and 10p for children.
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.
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?
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
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. . . .
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.
This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
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?
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.
What are the last two digits of 2^(2^2003)?
Can you work out how many of each kind of pencil this student
What is the last digit of the number 1 / 5^903 ?
How many six digit numbers are there which DO NOT contain a 5?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?