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Algebraic number thoery

Problem 1: Prove that there are no integers x, y, and z such that
x^2 +y^2 + z^2 = 999

Problem 2: Show that square root of 2 cubed is an irrational number.

Problem 3: For each of the following pairs a and b, use the division algorithm to find quotient q and remainder r.
(a) b=189, a=17
(b) b=111, a= -5
(c) b= -1029, a=27
(d) b= -111, a= -4

Solution Preview

Problem #1
Proof: Since 999 is an odd number, if there exists integers x,y,z such that
x^2+y^2+z^2=999, then we only have two cases.
Case 1: x,y,z are all odd numbers. An odd number has the form 2k+1 and thus
(2k+1)^2=4k(k+1)+1=1 (mod 8). So x^2+y^2+z^2=1+1+1=3 (mod 8). But
999=7 (mod 8), we get ...

Solution Summary

There are three problems here: a proof regarding sum of squared numbers, a proof of irrationality, and a set of division algorithm problems.

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