# 7.6: Exercises

• Bob Dumas and John E. McCarthy
• University of Washington and Washington University in St. Louis

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EXERCISE 7.1. Let $$n \in \mathbb{N}$$. Prove that if $$n$$ is not prime then $$n$$ has a prime factor $$p \leq \sqrt{n}$$.

EXERCISE $$7.2$$. Are $$15,462,227$$ and $$15,462,229$$ relatively prime?

EXERCISE 7.3. If $$n \in \mathbb{N}$$, under what conditions are $$n$$ and $$n+2$$ relatively prime?

EXERCISE 7.4. Prove that every natural number may be written as the product of a power of 2 and an odd number.

EXERCISE 7.5. Find $$\operatorname{gcd}(8243235,453169)$$.

EXERCISE 7.6. Find $$\operatorname{gcd}(15570555,10872579)$$.

EXERCISE 7.7. Let $$a$$ and $$b$$ be integers and $$m=\operatorname{gcd}(a, b)$$. Prove that $$\frac{a}{m}$$ and $$\frac{b}{m}$$ are relatively prime integers.

EXERCISE 7.8. Let $$a$$ and $$b$$ be positive integers with prime decomposition given by $a=\prod_{n=1}^{N} p_{n}^{r_{n}}$ and $b=\prod_{n=1}^{N} p_{n}^{s_{n}}$ where $$p_{n}, r_{n}, s_{n} \in \mathbb{N}$$ and $$p_{n}$$ is prime for $$1 \leq n \leq N$$. Prove that if $$t_{n}=\min \left(r_{n}, s_{n}\right)$$ for $$1 \leq n \leq N$$, then $\operatorname{gcd}(a, b)=\prod_{n=1}^{N} p_{n}^{t_{n}} .$ EXERCISE 7.9. In the statement of Lemma 7.14, suppose that $$\operatorname{gcd}(a, n) \neq$$ 1. Prove that the function $$\phi_{a}$$ is not a permutation of $$\mathbb{Z}_{n}^{*}$$. EXERCISE 7.10. Prove Proposition 7.16.

EXERCISE 7.11. Is 4757 prime?

EXERCISE 7.12. Define an ideal of $$\mathbb{Z}$$ in the natural way: A set $$I \subseteq \mathbb{Z}$$ is an ideal of $$\mathbb{Z}$$ if for any $$m, n \in I$$ and $$c \in \mathbb{Z}$$,

1. $$m+n \in I$$

and

1. $$m c \in I$$.

If $$a, b \in \mathbb{Z}$$, prove that the set of integer combinations of $$a$$ and $$b$$ are an ideal of $$\mathbb{Z}$$.

EXERCISE 7.13. Prove that every ideal of $$\mathbb{Z}$$ is principal. (Hint: find the generator of the ideal.)

EXERCISE 7.14. Let $$p$$ be prime and $$\mathbb{Z}_{p}[x]$$ be the set of polynomials with coefficients in $$\mathbb{Z}_{p}$$. What can you say about the roots of the polynomial $$x^{p-1}-[1]$$ in $$\mathbb{Z}_{p}$$ ? (We say that $$[a] \in \mathbb{Z}_{p}$$ is a root of a polynomial $$f \in \mathbb{Z}_{p}[x]$$ if $$\left.f([a])=[0] .\right)$$

EXERCISE 7.15. Prove that 0 is the additive identity in $$\mathbb{R}[x]$$ and 1 is the multiplicative identity in $$\mathbb{R}[x]$$. Use the formal definitions of addition and multiplication in $$\mathbb{R}[x]$$.

EXERCISE 7.16. Prove that the degree of the product of polynomials is equal to the sum of the degrees of the polynomials. Use the formal definition of multiplication in $$\mathbb{R}[x]$$.

EXERCISE 7.17. Let $$p \in \mathbb{R}[x]$$. Prove that $$p$$ has an additive inverse in $$\mathbb{R}[x]$$. Prove that $$p$$ has a multiplicative inverse iff $$p$$ has degree 0 . Use the formal definitions of addition and multiplication in $$\mathbb{R}[x]$$.

EXERCISE 7.18. Prove that addition and multiplication in $$\mathbb{R}[x]$$ are associative and commutative, and that the distributive property holds. Use the formal definitions of addition and multiplication in $$\mathbb{R}[x]$$.

EXERCISE 7.19. For $$0 \leq n \leq N$$, let $$a_{n} \in \mathbb{R}$$. If $$f=\sum_{n=0}^{N} a_{n} x^{n}$$ and $$g(x)=x-1$$, find the unique quotient and remainder where $$f$$ is the dividend and $$g$$ is the divisor. EXERCISE 7.20. Let $$f, g, q \in \mathbb{R}[x], g \neq 0$$. Suppose that $$f$$ is the dividend, $$g$$ the divisor and $$q$$ the quotient. Prove that the sum of the degree of $$g$$ and the degree of $$q$$ equals the degree of $$f$$.

EXERCISE 7.21. Is there a version of the Euclidean Algorithm for $$\mathbb{R}[x]$$ ?

This page titled 7.6: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Bob Dumas and John E. McCarthy via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.