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

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Adding All Nine

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

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14 Divisors

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

What is the smallest number with exactly 14 divisors?

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A First Product Sudoku

Stage: 3 Challenge Level: Challenge Level:1

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

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Three Times Seven

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Factors and Multiple Challenges

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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X Marks the Spot

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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" .

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Adding in Rows

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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LCM Sudoku

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Factoring a Million

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Divisively So

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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American Billions

Stage: 3 Challenge Level: Challenge Level:1

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...

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Even So

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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LCM Sudoku II

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

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

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Shifting Times Tables

Stage: 3 Challenge Level: Challenge Level:1

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

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Shifting Times Tables Game

Stage: 2 and 3 Challenge Level: Challenge Level:1

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

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What Numbers Can We Make?

Stage: 3 Challenge Level: Challenge Level:1

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

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Multiplication Equation Sudoku

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Product Sudoku

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

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

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GOT IT

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Cuboids

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Ben's Game

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Funny Factorisation

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

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Product Sudoku 2

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Dozens

Stage: 3 Challenge Level: Challenge Level:1

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

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Repeaters

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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How Old Are the Children?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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?"

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GOT IT Now

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Reverse to Order

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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Special Sums and Products

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Thirty Six Exactly

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Inclusion Exclusion

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Can You Find a Perfect Number?

Stage: 2 and 3

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

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Eminit

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Helen's Conjecture

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Hidden Squares

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Big Powers

Stage: 3 and 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Oh! Hidden Inside?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Transposition Cipher

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Sieve of Eratosthenes

Stage: 3 Challenge Level: Challenge Level:1

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

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Factoring Factorials

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Missing Multipliers

Stage: 3 Challenge Level: Challenge Level:1

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

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The Remainders Game

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge Level:2

A game that tests your understanding of remainders.

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Mathematical Swimmer

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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Counting Cogs

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Data Chunks

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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Different by One

Stage: 4 Challenge Level: Challenge Level:1

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

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Charlie's Delightful Machine

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Factors and Multiples Game

Stage: 2 and 3 Challenge Level: Challenge Level:1

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

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Squaresearch

Stage: 4 Challenge Level: Challenge Level:1

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

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What a Joke

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?


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