Monthly Archives: October 2011

Number Theory Notes (22 Oct 2011) – Part III

Posted on October 28, 2011 by limsup

Finally, we shall solve two more problems – the last problem is rather surprising since at first glance, it doesn’t appear to involve congruences. Problem 4 : Prove that if n is a perfect square, then . Solution : this is rather … Continue reading →

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Number Theory Notes (22 Oct 2011) – Part II

Posted on October 28, 2011 by limsup

Now we will solve actual problems with the theory we’ve just learnt. Problem 1 : Find all integers x such that x ÷ 5 has remainder 3, x ÷ 7 has remainder 6 and x ÷ 9 has remainder 2. Solution : This can be … Continue reading →

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Number Theory Notes (22 Oct 2011) – Part I

Posted on October 28, 2011 by limsup

Background required: none. For the first lecture, we shall look at congruence and modular arithmetic. Many of you may have already known (at least on an intuitive level) that square integers can only end in 0, 1, 4, 5, 6, … Continue reading →

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Homework (22 Oct 2011)

Posted on October 27, 2011 by limsup

Here’re the problems: A square integer N ends in 4 identical digits d in its decimal representation, where . Find all possible values of d. For each admissable value of d, find a possible N. N is a perfect square whose second-to-last … Continue reading →

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