Resources tagged with: Divisibility

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Broad Topics > Properties of Numbers > Divisibility

396

Age 14 to 16
Challenge Level

The four digits 5, 6, 7 and 8 are put at random in the spaces of the number : 3 _ 1 _ 4 _ 0 _ 9 2 Calculate the probability that the answer will be a multiple of 396.

What Numbers Can We Make Now?

Age 11 to 14
Challenge Level

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

Oh! Hidden Inside?

Age 11 to 14
Challenge Level

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

Odd Stones

Age 14 to 16
Challenge Level

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.

Take Three from Five

Age 14 to 16
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?

Mod 3

Age 14 to 16
Challenge Level

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

Novemberish

Age 14 to 16
Challenge Level

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.

Eminit

Age 11 to 14
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?

Check Codes

Age 14 to 16
Challenge Level

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check. . . .

Transposition Fix

Age 14 to 16
Challenge Level

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine. . . .

Powerful Factorial

Age 11 to 14
Challenge Level

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!?

Divisibility Tests

Age 11 to 16

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

Fac-finding

Age 14 to 16
Challenge Level

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.

What an Odd Fact(or)

Age 11 to 14
Challenge Level

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

Knapsack

Age 14 to 16
Challenge Level

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.

Elevenses

Age 11 to 14
Challenge Level

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

Power Mad!

Age 11 to 14
Challenge Level

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

Multiple Surprises

Age 11 to 16
Challenge Level

Sequences of multiples keep cropping up...

Latin Numbers

Age 14 to 16
Challenge Level

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

Going Round in Circles

Age 11 to 14
Challenge Level

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

What Numbers Can We Make?

Age 11 to 14
Challenge Level

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

Flow Chart

Age 11 to 14
Challenge Level

The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.

Remainders

Age 7 to 14
Challenge Level

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

Expenses

Age 14 to 16
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?

Digital Roots

Age 7 to 14

In this article for teachers, Bernard Bagnall describes how to find digital roots and suggests that they can be worth exploring when confronted by a sequence of numbers.

Dozens

Age 7 to 14
Challenge Level

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Square Routes

Age 11 to 14
Challenge Level

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?

Differences

Age 11 to 14
Challenge Level

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

AB Search

Age 11 to 14 Short
Challenge Level

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

Digat

Age 11 to 14
Challenge Level

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

Gaxinta

Age 11 to 14
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?

Squaresearch

Age 14 to 16
Challenge Level

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

Factoring Factorials

Age 11 to 14
Challenge Level

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

Divisively So

Age 11 to 14
Challenge Level

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

Remainder

Age 11 to 14
Challenge Level

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Peaches Today, Peaches Tomorrow...

Age 11 to 14
Challenge Level

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?

N000ughty Thoughts

Age 14 to 16
Challenge Level

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

The Remainders Game

Age 7 to 14
Challenge Level

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

Counting Factors

Age 11 to 14
Challenge Level

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

Just Repeat

Age 11 to 14
Challenge Level

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

LCM Sudoku

Age 14 to 16
Challenge Level

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

Repeaters

Age 11 to 14
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.

There's Always One Isn't There

Age 14 to 16
Challenge Level

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

Three Times Seven

Age 11 to 14
Challenge Level

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Big Powers

Age 11 to 16
Challenge Level

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.

Legs Eleven

Age 11 to 14
Challenge Level

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?

Ben's Game

Age 11 to 16
Challenge Level

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

Obviously?

Age 14 to 18
Challenge Level

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

Why 24?

Age 14 to 16
Challenge Level

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.

The Chinese Remainder Theorem

Age 14 to 18

In this article we shall consider how to solve problems such as "Find all integers that leave a remainder of 1 when divided by 2, 3, and 5."