# Factors, Multiples and Primes

### Counting Cogs

##### KS 2 & 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?

### Factors and Multiples Game

##### KS 2, 3 & 4 Challenge Level:

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

### Multiple Surprises

##### KS 3 Challenge Level:

Sequences of multiples keep cropping up...

### How Much Can We Spend?

##### KS 3 Challenge Level:

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

### Gabriel's Problem

##### KS 3 Challenge Level:

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

### Reversible Primes

##### KS 3 Short Challenge Level:

How many two-digit primes are there between 10 and 99 which are also prime when reversed?

### Product Sudoku

##### KS 3 Challenge Level:

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

### Factors and Multiples Puzzle

##### KS 3 Challenge Level:

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

### What Numbers Can We Make?

##### KS 3 Challenge Level:

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

### Divisible Digits

##### KS 3 Short Challenge Level:

Can you find the missing digits, given that the number is divisible by 3, 4, 5 and 6?

### American Billions

##### KS 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...

### Sieve of Eratosthenes

##### KS 3 Challenge Level:

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

### Dozens

##### KS 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?

### Stars

##### KS 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?

### Diminishing Returns

##### KS 3 Challenge Level:

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

### Times and Square

##### KS 3 Short Challenge Level:

Roger multiplies two consecutive integers and squares the result. Can you find the last two digits of his new number?

### Counting Factors

##### KS 3 Challenge Level:

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

### Red Card Blue Card

##### KS 3 Short Challenge Level:

Can you arrange the red and blue cards so that the rules are all followed?

### Route to Infinity

##### KS 3 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next?

### Eight Factors Only

##### KS 3 Short Challenge Level:

We are given two factors of a number with eight factors. Can you work out the other factors of the number?

##### KS 3 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

### Colossal Sum

##### KS 3 Short Challenge Level:

What is the units digit in this sum of powers of 9?

### Differences

##### KS 3 Challenge Level:

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

### Cancelling Fractions

##### KS 3 Short Challenge Level:

Can you find a fraction with the following properties?

### Leap Monday

##### KS 3 Short Challenge Level:

Can you find the next time that the 29th of February will fall on a Monday?

### Funny Factorisation

##### KS 3 Challenge Level:

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 you find the factors?

### Big Blackboard

##### KS 3 Short Challenge Level:

Can you work out which numbers between 1 and 2016 have exactly two of 2, 3, 4 as factors?

### Divisibility Tests

##### KS 3 & 4

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

### Beelines

##### KS 4 Challenge Level:

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

### Take Three from Five

##### KS 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?

### Expenses

##### KS 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?

### Filling the Gaps

##### KS 4 Challenge Level:

Which numbers can we write as a sum of square numbers?

### Adding a Square to a Cube

##### KS 4 Short Challenge Level:

If you take a number and add its square to its cube, how often will you get a perfect square?