Homework 20

Read: End of Section 4.4 and Section 6.2. Also read Theorem 6.17 in Section 6.1

Turn in: 4.36, 4.38, 6.19, and the additional problem below

Additional Problem: Let $a \in \mathbb{Z}$ with $a\ge 2$. Prove that if $a$ is a square (i.e. $a = b^2$ for some $b\in \mathbb{Z}$), then the number of times $p$ appears in the prime factorization of $a$ is an even number.

• There is a hint for 4.38 in the paragraph before it, but even so, it’s a bit challenging.
• Please do the Additional Problem below before doing 6.19, and please read Theorem 6.17 in Section 6.1, before doing the Additional Problem.
• 6.19 has a hint in the paragraph before it, but I think it’s hard to follow. You will definitely want to do a proof by contradiction. Assume $\sqrt{2}$ is a rational number, so $\sqrt{2} = \frac{m}{n}$ for some $m,n\in \mathbb{Z}$. Squaring and rearranging, you get $2n^2 = m^2$. Now find a contradiction to Theorem 6.17 using the Additional Problem.