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There are **232** NRICH Mathematical resources connected to **Factors and multiples**, you may find related items under Properties of numbers.

Problem
Primary curriculum
Secondary curriculum
### Stars

Can you work out what step size to take to ensure you visit all the dots on the circle?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cuboids

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ben's Game

Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lots of Lollies

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Share Bears

Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Mixed-up Clock

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Take Three from Five

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

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fac-finding

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fitted

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Which Is Quicker?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Remainders

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Age 7 to 14

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Telescoping Functions

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Divisibility Tests

This article explains various divisibility rules and why they work. An article to read with pencil and paper handy.

Age 11 to 16

Problem
Primary curriculum
Secondary curriculum
### Got It

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.

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Primes Make One Square

Can you make square numbers by adding two prime numbers together?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factor Lines

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

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shopping Basket

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Multiplication Squares

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fractions in a Box

The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mystery Matrix

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Moons of Vuvv

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Flashing Lights

Norrie sees two lights flash at the same time, then one of them flashes every 4th second, and the other flashes every 5th second. How many times do they flash together during a whole minute?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Abundant Numbers

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Zios and Zepts

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Heads and Feet

On a farm there were some hens and sheep. Altogether there were 8 heads and 22 feet. How many hens were there?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Rules - OK

Can you produce convincing arguments that a selection of statements about numbers are true?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### American Billions

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...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One to Eight

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Why 24?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Funny Factorisation

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Beelines

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Neighbours

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Expenses

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Big Powers

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.

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summing Consecutive Numbers

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Counting Factors

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Have You Got It?

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Common Divisor

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Em'power'ed

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Grouping Goodies

Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Sticks

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Biscuit Decorations

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Round and Round the Circle

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sweets in a Box

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pebbles

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?

Age 7 to 11

Challenge Level