User:Marty/UCSC Math 110 Fall 2008/Homework 2

This is Homework 2, due on Wednesday, October 8, 2008, at 9:30 am. Show all of your work.

$$35x + 55y + 77z = 4.$$
 * Problem 1, 2 points: Use the Euclidean algorithm (keep your work neat and organized) to find three integers $x,y,z$ such that:


 * Problem 2, 3 points: Division with remainder also works for polynomials with rational or real coefficients.  Namely, if $A(x)$ and $B(x)$ are two (nonzero) polynomials with coefficients in $\RR$, then there exist polynomials $Q(x)$ and $R(x)$, such that $A(x) = Q(x) B(x) + R(x)$, and the degree of $R(x)$ is less than the degree of $B(x)$.

Use the Euclidean algorithm, to find the greatest common divisor of the polynomials: $$A(x) = x^7-2x^5+x^2-3x^3+1, \mbox{ and } B(x) = x^5+3x^3+2x^2+2x+2.$$


 * Problem 3, 3 points: Suppose that $n \in \NN$, and $p$ is a prime number.  Prove that if $p$ divides $n^2$, then $p^2$ divides $n^2$.


 * Problem 4, 2 points: Play in your sandbox as follows:
 * Click on your "sandbox" page (within your user page), to create it, if you haven't already done so.
 * Copy and paste the text in the box below into your sandbox page.
 * Change all three parts of the text in your sandbox page, so that the resulting mathematical statements are true.

This is Problem 4 of Homework 2
The first statement is:
 * $ \frac{2}{3} + \frac{3}{4} = \frac{5}{9}. $

The second statement is:
 * If $x$ is an integer, then $x^{2 + 3} = (x^2)^3$.

The third statement is:
 * If $n \in \NN$, and $n \geq 1$, then $\sum_{i=1}^{n+3} i = i^2$.