There are 70 NRICH Mathematical resources connected to Divisibility, you may find related items under Properties of numbers.
Broad Topics > Properties of numbers > Divisibility
You'll need to know your number properties to win a game of Statement Snap...
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Play this game and see if you can figure out the computer's chosen number.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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.
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over?
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?
Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?
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?
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.
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
This article explains various divisibility rules and why they work. An article to read with pencil and paper handy.
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...
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.
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?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
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?
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.
Is there an efficient way to work out how many factors a large number has?
Can you create a Latin Square from multiples of a six digit number?
What can you say about the common difference of an AP where every term is prime?
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!?
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?
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?
Find the highest power of 11 that will divide into 1000! exactly.
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?
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?
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.
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?
Find the number which has 8 divisors, such that the product of the divisors is 331776.
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.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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.
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.
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
How many noughts are at the end of these giant numbers?
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.
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!