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

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. . . .

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?

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?

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.

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?

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.

Hidden Rectangles

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?

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. . . .

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?

A Biggy

Stage: 4 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.

Sixational

Stage: 4 and 5 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

For What?

Stage: 4 Challenge Level:

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Common Divisor

Stage: 4 Challenge Level:

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

What Numbers Can We Make Now?

Stage: 3 Challenge Level:

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

Number Rules - OK

Stage: 4 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Diagonal Product Sudoku

Stage: 3 and 4 Challenge Level:

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

Transposition Cipher

Stage: 3 and 4 Challenge Level:

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

Star Product Sudoku

Stage: 3 and 4 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.

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?

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)

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.

LCM Sudoku II

Stage: 3, 4 and 5 Challenge Level:

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

Mod 3

Stage: 4 Challenge Level:

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

Helen's Conjecture

Stage: 3 Challenge Level:

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

N000ughty Thoughts

Stage: 4 Challenge Level:

How many noughts are at the end of these giant numbers?

Have You Got It?

Stage: 3 Challenge Level:

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

LCM Sudoku

Stage: 4 Challenge Level:

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

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?

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?

Factor Lines

Stage: 2 and 3 Challenge Level:

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

Counting Factors

Stage: 3 Challenge Level:

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

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?

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?

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.

Squaresearch

Stage: 4 Challenge Level:

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

Really Mr. Bond

Stage: 4 Challenge Level:

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Substitution Transposed

Stage: 3 and 4 Challenge Level:

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

One to Eight

Stage: 3 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.

Expenses

Stage: 4 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?

Remainder

Stage: 3 Challenge Level:

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

Take Three from Five

Stage: 4 Challenge Level:

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?

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?

What a Joke

Stage: 4 Challenge Level:

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

Factoring Factorials

Stage: 3 Challenge Level:

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

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?

Shifting Times Tables

Stage: 3 Challenge Level:

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

Factor Track

Stage: 2 and 3 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.

Sieve of Eratosthenes

Stage: 3 Challenge Level:

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