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Monthly Archives: October 2011
Number Theory Notes (22 Oct 2011) – Part III
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
Number Theory Notes (22 Oct 2011) – Part II
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
Number Theory Notes (22 Oct 2011) – Part I
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
Homework (22 Oct 2011)
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 secondtolast … Continue reading