Factors and multiples

problem
Cogs

Age

11 to 14

Challenge level

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the cogwheel A as the wheels rotate.
problem
Even so

Age

11 to 14

Challenge level

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?
problem
Factoring a million

Age

14 to 16

Challenge level

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
problem
American billions

Age

11 to 14

Challenge level

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...
problem
Hot pursuit

Age

11 to 14

Challenge level

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...
problem
One to eight

Age

11 to 14

Challenge level

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.
problem
Adding in rows

Age

11 to 14

Challenge level

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?
problem
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.
problem
Funny factorisation

Age

11 to 16

Challenge level

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

Age

11 to 14

Challenge level

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