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

### Crossings

##### Stage: 2 Challenge Level:

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

### 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.

### 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?

##### 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?

### Have You Got It?

##### Stage: 3 Challenge Level:

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

### 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.

### A First Product Sudoku

##### Stage: 3 Challenge Level:

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

### What Do You Need?

##### Stage: 2 Challenge Level:

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

### 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?

### How Old Are the Children?

##### Stage: 3 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?"

### 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?

### Three Neighbours

##### Stage: 2 Challenge Level:

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

### 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.

### 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?

### 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?

### 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 Dice

##### Stage: 2 Challenge Level:

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

### 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.

### American Billions

##### Stage: 3 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...

### Fitted

##### Stage: 2 Challenge Level:

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

### Gaxinta

##### Stage: 3 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?

### Colour Wheels

##### Stage: 2 Challenge Level:

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

### 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.

### Down to Nothing

##### Stage: 2 Challenge Level:

A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.

### Oh! Hidden Inside?

##### Stage: 3 Challenge Level:

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

### Dozens

##### Stage: 2 and 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?

##### 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. . . .

### The Moons of Vuvv

##### Stage: 2 Challenge Level:

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

### Mystery Matrix

##### Stage: 2 Challenge Level:

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

### Diagonal Product Sudoku

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

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

### AB Search

##### Stage: 3 Challenge Level:

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

### Counting Factors

##### Stage: 3 Challenge Level:

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

### Money Measure

##### Stage: 2 Challenge Level:

How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?

### Multiplication Series: Number Arrays

##### Stage: 1 and 2

This article for teachers describes how number arrays can be a useful reprentation for many number concepts.

### Music to My Ears

##### Stage: 2 Challenge Level:

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

### Curious Number

##### Stage: 2 Challenge Level:

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

### It Figures

##### Stage: 2 Challenge Level:

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Got it for Two

##### Stage: 2 Challenge Level:

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

### Gabriel's Problem

##### Stage: 3 Challenge Level:

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

### Give Me Four Clues

##### Stage: 2 Challenge Level:

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

### Remainders

##### Stage: 3 Challenge Level:

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?

### Times Tables Shifts

##### Stage: 2 Challenge Level:

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

### Product Sudoku

##### Stage: 3 Challenge Level:

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

### 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?

### 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.

### The Remainders Game

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

A game that tests your understanding of remainders.

### Neighbours

##### Stage: 2 Challenge Level:

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

### Round and Round the Circle

##### Stage: 2 Challenge Level:

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

### Eminit

##### Stage: 3 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?

### Multiplication Squares

##### Stage: 2 Challenge Level:

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.