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# STEP on Up

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Square Difference

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Divisible Factorisations

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Proper Factors

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Square Remainders

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In this feature there are 4 problems based on STEP questions, all of which involve number theory. You can find more number theory questions here.

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.

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?

Age 16 to 18

Challenge Level

Can you show that $n^5-n^3$ is always divisible by $24$?

Age 16 to 18

Challenge Level

Can you find the smallest integer which has exactly 426 proper factors?

Age 16 to 18

Challenge Level

What is the remainder if you divide a square number by $8$?