Factors and multiples

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
Have you got it?

Age

11 to 14

Challenge level

Can you explain the strategy for winning this game with any target?
problem
Sixational

Age

14 to 18

Challenge level

The nth term of a sequence is given by the formula n^3 + 11n. Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.
problem
A biggy

Age

14 to 16

Challenge level

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
problem
Common divisor

Age

14 to 16

Challenge level

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
problem
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.
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
N000ughty thoughts

Age

14 to 16

Challenge level

How many noughts are at the end of these giant numbers?
problem
Em'power'ed

Age

16 to 18

Challenge level

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?
problem
Some cubes

Age

16 to 18

Challenge level

The sum of the cubes of two numbers is 7163. What are these numbers?