Factors and multiples

problem
Beelines

Age

14 to 16

Challenge level

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
problem
Expenses

Age

14 to 16

Challenge level

What is the largest number which, when divided into these five numbers in turn, leaves the same remainder each time?
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
Number rules - OK

Age

14 to 16

Challenge level

Can you produce convincing arguments that a selection of statements about numbers are true?
problem
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.
problem
Filling the gaps

Age

14 to 16

Challenge level

Which numbers can we write as a sum of square numbers?
problem
Latin numbers

Age

14 to 16

Challenge level

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

Age

16 to 18

Challenge level

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
problem
Powerful factors

Age

16 to 18

Challenge level

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.
problem
Polite numbers

Age

16 to 18

Challenge level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?