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

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AB Search

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

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

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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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Powerful Factorial

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

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!?

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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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Gaxinta

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

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?

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Ewa's Eggs

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

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

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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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Remainders

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

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

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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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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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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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Times Right

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

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?

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Digat

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

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

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Remainder

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

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

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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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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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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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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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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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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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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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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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Hot Pursuit

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

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?

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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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Phew I'm Factored

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

Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

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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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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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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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Diggits

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

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

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Expenses

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

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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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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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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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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Two Much

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

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

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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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One to Eight

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

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.

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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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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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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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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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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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Shopping Basket

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

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

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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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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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Substitution Transposed

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

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

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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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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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Take Three from Five

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Pebbles

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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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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