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There are **70** NRICH Mathematical resources connected to **Divisibility**, you may find related items under Properties of numbers.

Problem
Primary curriculum
Secondary curriculum
### Statement Snap

You'll need to know your number properties to win a game of Statement Snap...

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Division Rules

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What Numbers Can We Make Now?

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What Numbers Can We Make?

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Curious Number

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Going Round in Circles

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Age 11 to 14

Challenge Level

Game
Primary curriculum
Secondary curriculum
### The Remainders Game

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

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Mad!

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Elevenses

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### LCM Sudoku

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Odd Stones

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

Age 14 to 16

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
### 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
### Peaches Today, Peaches Tomorrow...

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What an Odd Fact(or)

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

Age 11 to 14

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
### 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
### 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
### 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
### 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
### 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
### Differences

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

Age 11 to 14

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
### Legs Eleven

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

Age 11 to 14

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
### 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
### Prime AP

What can you say about the common difference of an AP where every term is prime?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Powerful Factorial

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!?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Eminit

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Adding in Rows

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factoring Factorials

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Book Codes

Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gaxinta

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Digat

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### AB Search

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

Age 11 to 14

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Skeleton

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Routes

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Oh! Hidden Inside?

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Repeaters

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squaresearch

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sixational

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. Prove that all terms of the sequence are divisible by 6.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mod 3

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### N000ughty Thoughts

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Adding All Nine

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 other possibilities for yourself!

Age 11 to 14

Challenge Level