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Broad Topics > Numbers and the Number System > Factors and multiples

Power Crazy

Stage: 3 Challenge Level:

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?

Factors and Multiples Game

Stage: 2 and 3 Challenge Level:

A game in which players take it in turns to choose a number. Can you block your opponent?

Charlie's Delightful Machine

Stage: 3 and 4 Challenge Level:

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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?

Stars

Stage: 3 Challenge Level:

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

The Remainders Game

Stage: 2 and 3 Challenge Level:

A game that tests your understanding of remainders.

Hidden Squares

Stage: 3 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Factoring a Million

Stage: 4 Challenge Level:

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

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?

Big Powers

Stage: 3 and 4 Challenge Level:

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

LCM Sudoku

Stage: 4 Challenge Level:

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

Data Chunks

Stage: 4 Challenge Level:

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .

Stage: 3 Challenge Level:

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Stage: 3 Challenge Level:

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

Three Times Seven

Stage: 3 Challenge Level:

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Factors and Multiple Challenges

Stage: 3 Challenge Level:

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. . . .

Remainder

Stage: 3 Challenge Level:

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Cuboids

Stage: 3 Challenge Level:

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Factors and Multiples - Secondary Resources

Stage: 3 and 4 Challenge Level:

A collection of resources to support work on Factors and Multiples at Secondary level.

How Old Are the Children?

Stage: 3 Challenge Level:

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

Multiplication Equation Sudoku

Stage: 4 and 5 Challenge Level:

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

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?

14 Divisors

Stage: 3 Challenge Level:

What is the smallest number with exactly 14 divisors?

Repeaters

Stage: 3 Challenge Level:

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

What Numbers Can We Make?

Stage: 3 Challenge Level:

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

GOT IT Now

Stage: 2 and 3 Challenge Level:

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

Squaresearch

Stage: 4 Challenge Level:

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

Dozens

Stage: 3 Challenge Level:

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Special Sums and Products

Stage: 3 Challenge Level:

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Product Sudoku 2

Stage: 3 and 4 Challenge Level:

Given the products of diagonally opposite cells - can you complete this Sudoku?

Got It

Stage: 2 and 3 Challenge Level:

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Shifting Times Tables

Stage: 3 Challenge Level:

Can you find a way to identify times tables after they have been shifted up?

Diggits

Stage: 3 Challenge Level:

Can you find what the last two digits of the number $4^{1999}$ are?

American Billions

Stage: 3 Challenge Level:

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Inclusion Exclusion

Stage: 3 Challenge Level:

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

Eminit

Stage: 3 Challenge Level:

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Hot Pursuit

Stage: 3 Challenge Level:

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

Different by One

Stage: 4 Challenge Level:

Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)

Missing Multipliers

Stage: 3 Challenge Level:

What is the smallest number of answers you need to reveal in order to work out the missing headers?

Exploring Simple Mappings

Stage: 3 Challenge Level:

Explore the relationship between simple linear functions and their graphs.

Sieve of Eratosthenes

Stage: 3 Challenge Level:

Follow this recipe for sieving numbers and see what interesting patterns emerge.

Transposition Cipher

Stage: 3 and 4 Challenge Level:

Can you work out what size grid you need to read our secret message?

What Numbers Can We Make Now?

Stage: 3 and 4 Challenge Level:

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Substitution Transposed

Stage: 3 and 4 Challenge Level:

Substitution and Transposition all in one! How fiendish can these codes get?

Counting Cogs

Stage: 2 and 3 Challenge Level:

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

Mathematical Swimmer

Stage: 3 Challenge Level:

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths. . . .

Ben's Game

Stage: 3 Challenge Level:

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Factoring Factorials

Stage: 3 Challenge Level:

Find the highest power of 11 that will divide into 1000! exactly.

Product Sudoku

Stage: 3, 4 and 5 Challenge Level:

The clues for this Sudoku are the product of the numbers in adjacent squares.
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