Resources tagged with: Factors and multiples

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

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?

Dozens

Age 7 to 14 Challenge Level:

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

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?

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?

Expenses

Age 14 to 16 Challenge Level:

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

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?

Squaresearch

Age 14 to 16 Challenge Level:

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

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.

LCM Sudoku

Age 14 to 16 Challenge Level:

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

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...

Divisibility Tests

Age 11 to 16

This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

Two Much

Age 11 to 14 Challenge Level:

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

Factoring Factorials

Age 11 to 14 Challenge Level:

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

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?

Ewa's Eggs

Age 11 to 14 Challenge Level:

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?

Digat

Age 11 to 14 Challenge Level:

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

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!?

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?

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.

Factor Track

Age 7 to 14 Challenge Level:

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

Counting Factors

Age 11 to 14 Challenge Level:

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

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?

Can You Find a Perfect Number?

Age 7 to 14

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

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?

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?

Mod 3

Age 14 to 16 Challenge Level:

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Different by One

Age 14 to 16 Challenge Level:

Can you make lines of Cuisenaire rods that differ by 1?

Factoring a Million

Age 14 to 16 Challenge Level:

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

The Remainders Game

Age 7 to 14 Challenge Level:

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

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?

Satisfying Statements

Age 11 to 14 Challenge Level:

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

A Biggy

Age 14 to 16 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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?

Transposition Cipher

Age 11 to 16 Challenge Level:

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

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.

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?

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?

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?

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.

Mathematical Swimmer

Age 11 to 14 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 I. . . .

Product Sudoku

Age 11 to 16 Challenge Level:

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

What a Joke

Age 14 to 16 Challenge Level:

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

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?

Inclusion Exclusion

Age 11 to 14 Challenge Level:

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

Sieve of Eratosthenes

Age 11 to 14 Challenge Level:

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

Data Chunks

Age 14 to 16 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. . . .

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?

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.

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...

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?