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STEP is a mathematics admissions test taken at the end of Year 13 (June before starting university in September). It is compulsory for Mathematics places at Cambridge, and many other UK universities recommend students work through past papers, or give reduced offers for students who take STEP and do well in it.
For more information on STEP, and help in preparing for the tests, see the STEP Support Programme. The last day for sending in solutions to the live problems on this page is Monday 10th July 2023.
problem
Square difference
Age
16 to 18
Challenge level
Which numbers can you write as a difference of two squares? In how many ways can you write $pq$ as a difference of two squares if $p$ and $q$ are prime?
problem
Divisible factorisations
Age
16 to 18
Challenge level
Can you show that $n^5-n^3$ is always divisible by $24$?
problem
Proper factors
Age
16 to 18
Challenge level
Can you find the smallest integer which has exactly 426 proper factors?
problem
Square remainders
Age
16 to 18
Challenge level
What is the remainder if you divide a square number by $8$?