Factors and multiples

problem
Magical products

Age

11 to 14

Challenge level

Can you place the nine cards onto a 3x3 grid such that every row, column and diagonal has a product of 1?
problem
DOTS division

Age

14 to 16

Challenge level

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
problem
Squaresearch

Age

14 to 16

Challenge level

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

Age

14 to 16

Challenge level

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
problem
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.
problem
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?
problem
Inclusion exclusion

Age

11 to 14

Challenge level

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

Age

11 to 14

Challenge level

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
problem
Times right

Age

11 to 16

Challenge level

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?
problem
Factoring factorials

Age

11 to 14

Challenge level

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