All Secondary Factors and Multiples

game

Celebrate 25 years of NRICH

Age

5 to 14

To celebrate NRICH's 25th birthday, why not play this special version of our classic game, Got It? Can you devise a strategy so that you will always win?

interactivity

Cuisenaire environment

Age

5 to 16

Challenge level

An environment which simulates working with Cuisenaire rods.

article

Can you find a perfect number?

Can you find any perfect numbers? Read this article to find out more...

problem

Xavi's T-shirt

Age

7 to 16

Challenge level

How much can you read into a T-shirt?

problem

Missing multipliers

Age

7 to 14

Challenge level

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

problem

Dozens

Age

7 to 14

Challenge level

Can you select the missing digit(s) to find the largest multiple?

problem

Neighbourly addition

Age

7 to 14

Challenge level

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

game

Favourite

Factors and multiples game

Age

7 to 16

Challenge level

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

game

Favourite

The remainders game

Age

7 to 14

Challenge level

Play this game and see if you can figure out the computer's chosen number.

problem

3 by 3 Mathdokus

Age

7 to 14

Challenge level

Can you use the clues to complete these 3 by 3 Mathematical Sudokus?

problem

A first product Sudoku

Age

7 to 14

Challenge level

Given the products of adjacent cells, can you complete this Sudoku?

problem

4 by 4 Mathdokus

Age

7 to 14

Challenge level

Can you use the clues to complete these 4 by 4 Mathematical Sudokus?

game

Factors and multiples game for two

Age

7 to 14

Challenge level

Factors and Multiples game for an adult and child. How can you make sure you win this game?

problem

Statement snap

Age

7 to 14

Challenge level

You'll need to know your number properties to win a game of Statement Snap...

problem

5 by 5 Mathdokus

Age

7 to 16

Challenge level

Can you use the clues to complete these 5 by 5 Mathematical Sudokus?

problem

6 by 6 Mathdokus

Age

7 to 16

Challenge level

Can you use the clues to complete these 6 by 6 Mathematical Sudokus?

problem

Remainders

Age

7 to 14

Challenge level

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

problem

Counting cogs

Age

7 to 14

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?

problem

Three neighbours

Age

7 to 14

Challenge level

Take three consecutive numbers and add them together. What do you notice?

problem

Factor lines

Age

7 to 14

Challenge level

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

problem

Got it

Age

7 to 14

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.

$Got \It for Two$

game

Got it for two

Age

7 to 14

Challenge level

Got It game for an adult and child. How can you play so that you know you will always win?

article

Divisibility tests

This article explains various divisibility rules and why they work. An article to read with pencil and paper handy.

problem

LCM Sudoku II

Age

11 to 18

Challenge level

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

list

Factors and multiples - secondary resources

Age

11 to 16

Challenge level

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

problem

Multiples Sudoku

Age

11 to 14

Challenge level

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

problem

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 other possibilities for yourself!

problem

How much can we spend?

Age

11 to 14

Challenge level

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

problem

Shifting times tables

Age

11 to 14

Challenge level

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

problem

American billions

Age

11 to 14

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...

problem

Charlie's delightful machine

Age

11 to 16

Challenge level

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

problem

Product 100

Age

11 to 14

Challenge level

The product of four different positive integers is 100. What is the sum of these four integers?

problem

Have you got it?

Age

11 to 14

Challenge level

Can you explain the strategy for winning this game with any target?

problem

Factor trio

Age

11 to 14

Challenge level

Which of the numbers shown is the product of exactly 3 distinct prime factors?

problem

Summing consecutive numbers

Age

11 to 14

Challenge level

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

problem

What numbers can we make?

Age

11 to 14

Challenge level

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

problem

Diagonal product Sudoku

Age

11 to 16

Challenge level

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

problem

Sieve of Eratosthenes

Age

11 to 14

Challenge level

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

problem

Prime order

Age

11 to 14

Challenge level

Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

problem

Transposition cipher

Age

11 to 16

Challenge level

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

problem

Factors and multiples puzzle

Age

11 to 14

Challenge level

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

problem

Multiple surprises

Age

11 to 16

Challenge level

Sequences of multiples keep cropping up...

problem

Nine in a line

Age

11 to 14

Challenge level

The sum of 9 consecutive positive whole numbers is 2007. What is the largest of these numbers?

problem

Gabriel's problem

Age

11 to 14

Challenge level

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

problem

Number families

Age

11 to 14

Challenge level

How many different number families can you find?

problem

Flow chart

Age

11 to 14

Challenge level

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.

problem

Satisfying statements

Age

11 to 14

Challenge level

Can you find any two-digit numbers that satisfy all of these statements?

problem

How old are the children?

Age

11 to 14

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?"

problem

Power mad!

Age

11 to 14

Challenge level

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

problem

Funny factorisation

Age

11 to 16

Challenge level

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

problem

Take three from five

Age

11 to 16

Challenge level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

problem

Drawing Celtic knots

Age

11 to 14

Challenge level

Here is a chance to create some Celtic knots and explore the mathematics behind them.

problem

Four coloured lights

Age

11 to 14

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?

problem

Even so

Age

11 to 14

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?

problem

Cogs

Age

11 to 14

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 cogwheel A as the wheels rotate.

problem

Millennium leap

Age

11 to 14

Challenge level

How many leap years will there be between 2001 and 3001?

problem

Stars

Age

11 to 14

Challenge level

Can you work out what step size to take to ensure you visit all the dots on the circle?

problem

Adjacent factors

Age

11 to 14

Challenge level

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

problem

Great Granddad

Age

11 to 14

Challenge level

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

problem

Back of the queue

Age

11 to 14

Challenge level

Weekly Problem 48 - 2013
What is the remainder when the number 743589 × 301647 is divided by 5?

problem

Counting factors

Age

11 to 14

Challenge level

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

problem

Rolling along the trail

Age

11 to 14

Challenge level

What could be the scores from five throws of this dice?

problem

Power crazy

Age

11 to 14

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?

problem

Magical products

Age

11 to 14

Challenge level

Can you place the nine cards onto a 3x3 grid such that every row, column and diagonal has a product of 1?

problem

Repeaters

Age

11 to 14

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.

problem

Divisively so

Age

11 to 14

Challenge level

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

problem

Stair climb

Age

11 to 14

Challenge level

Three people run up stairs at different rates. If they each start from a different point - who will win, come second and come last?

problem

Product Sudoku

Age

11 to 16

Challenge level

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

problem

Oh! Hidden inside?

Age

11 to 14

Challenge level

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

problem

Remainder

Age

11 to 14

Challenge level

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

problem

Find from factors

Age

11 to 14

Challenge level

Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?

problem

What numbers can we make now?

Age

11 to 14

Challenge level

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

problem

AB search

Age

11 to 14

Challenge level

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

problem

Digat

Age

11 to 14

Challenge level

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

problem

Robotic rotations

Age

11 to 16

Challenge level

How did the the rotation robot make these patterns?

problem

Thirty six exactly

Age

11 to 14

Challenge level

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

problem

What does it all add up to?

Age

11 to 18

Challenge level

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

problem

Alison's quilt

Age

11 to 14

Challenge level

Nine squares are fitted together to form a rectangle. Can you find its dimensions?

problem

Star product Sudoku

Age

11 to 16

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.

problem

Trailing zeros

Age

11 to 14

Challenge level

How many zeros does 50! have at the end?

problem

Factoring factorials

Age

11 to 14

Challenge level

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

problem

Just repeat

Age

11 to 14

Challenge level

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

problem

Three times seven

Age

11 to 14

Challenge level

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

problem

Adding in rows

Age

11 to 14

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?

problem

One to eight

Age

11 to 14

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.

problem

Hot pursuit

Age

11 to 14

Challenge level

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

problem

Ben's game

Age

11 to 16

Challenge level

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

problem

Cuboids

Age

11 to 14

Challenge level

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

problem

Eminit

Age

11 to 14

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?

problem

Powerful factorial

Age

11 to 14

Challenge level

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?

problem

X marks the spot

Age

11 to 14

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" .

problem

Big powers

Age

11 to 16

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.

problem

Mathematical swimmer

Age

11 to 14

Challenge level

Can you work out how many lengths I swim each day?

problem

Substitution transposed

Age

11 to 16

Challenge level

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

problem

Differences

Age

11 to 14

Challenge level

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

problem

Shopping basket

Age

11 to 16

Challenge level

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

problem

Gaxinta

Age

11 to 14

Challenge level

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?