All Secondary Factors and Multiples

What is the smallest number of answers you need to reveal in order to work out the missing headers?

Given the products of adjacent cells, can you complete this Sudoku?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

Can you use the clues to complete these 3 by 3 Mathematical Sudokus?

Can you use the clues to complete these 4 by 4 Mathematical Sudokus?

Can you use the clues to complete these 5 by 5 Mathematical Sudokus?

Can you use the clues to complete these 6 by 6 Mathematical Sudokus?

Can you select the missing digit(s) to find the largest multiple?

How much can you read into a T-shirt?

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.

Take three consecutive numbers and add them together. What do you notice?

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

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

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

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

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

Given the products of diagonally opposite cells - can you complete this Sudoku?

Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

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!

Sequences of multiples keep cropping up...

The sum of 9 consecutive positive whole numbers is 2007. What is the largest of these numbers?

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

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...

How many different number families can you find?

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

Can you find any two-digit numbers that satisfy all of these statements?

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

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?

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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

Can you find a way to identify times tables after they have been shifted up or down?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

The product of four different positive integers is 100. What is the sum of these four integers?

Which of the numbers shown is the product of exactly 3 distinct prime factors?

Three people run up stairs at different rates. If they each start from a different point - who will win, come second and come last?

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

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

Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?

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

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

Nine squares are fitted together to form a rectangle. Can you find its dimensions?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

How did the the rotation robot make these patterns?

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

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

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

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?

Here is a chance to create some Celtic knots and explore the mathematics behind them.

Use properties of numbers to work out whether you can satisfy all these statements at the same time.

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

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

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Weekly Problem 48 - 2013
What is the remainder when the number 743589 × 301647 is divided by 5?

What could be the scores from five throws of this dice?

Can you place the nine cards onto a 3×3 grid such that every row, column and diagonal has a product of 1?

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

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

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?

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.

Weekly Problem 22 - 2008
The following sequence continues indefinitely... Which of these integers is a multiple of 81?

How many zeros does 50! have at the end?

6! = 6 × 5 × 4 × 3 × 2 × 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!?

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.

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

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

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

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?

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?

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?