1
A number such as 11, 111, 1111 is called a repunit. Clearly eleven is a prime repunit. Find two more, and say how you did it.
Section6.6Exercises¶ permalink
2
Find the prime numbers less than 100 using the Sieve of Eratosthenes (6.2.2). Make sure you actually draw it! Every math student should do this once, and only once.
3
Prove Lemma 6.3.5; if a prime \(p\) divides a product \(ab\), then \(p\) divides at least one of \(a\) or \(b\).
4
Prove Corollary 6.3.6; if a prime \(p\) divides any finite product of primes, then \(p\) divides at least one of them.
5
Prove that if \(\gcd(a,b)=1\) and \(a\mid bc\) then \(a\mid c\) as well, using the FTA.
6
Prove using the FTA that if \(\gcd(a,b)=d\) then \(\gcd\left(\frac{a}{d},\frac{b}{d}\right)=1\).
7
Assuming \(b \mid a\), find a formula to complete \begin{equation*}a/b=\prod_{i=1}^n p_i^{???}\end{equation*} and prove it using the FTA.
8
How would you describe a factorization of a rational number? Do you think you could extend the Fundamental Theorem of Arithmetic to this case? If so, how? If not, why would it not be appropriate?
9
Show that if \(a\) and \(b\) are positive integers and \(a^3\mid b^2\), then \(a\mid b\).
10
Show that if \(p^a\parallel m\) and \(p^b\parallel n\), then \(p^{a+b}\parallel mn\).
11
Prove Proposition 3.7.2 using the FTA; if \(\gcd(m,n)=1\) and \(mn\) is a perfect square, then so are \(m\) and \(n\).
12
Is it possible for \(n!\) to end in exactly five zeros?
13
Show that \(\log_{10} (5)\) is irrational.
14
Show that \(3^{2/3}\) is irrational.
15
By hand, find the prime factorizations of 36, 756, and 1001. Use these to find their pairwise gcds.
16
By hand, find the gcd of \(2^2\cdot 3^5\cdot 7^2\cdot 13\cdot 37\) and \(2^3\cdot 3^4\cdot 11\cdot 31^2\).
17
By any method you like, find the prime factorizations of \(2^{24}-1\) and \(10^8-1\), as well as their gcd.
18
Prove that the only solutions of \(x^2\equiv x\) (mod \(p\)) are \(x=[0]\) and \(x=[1]\), if \(p\) is a prime. (Refer to Question 4.6.7; this and the next exercise answer Exercise 4.7.14.)
19
Try to decide for exactly which composite moduli \(n\) the previous question is true. (Refer to the interact in Question 4.6.7; this and the previous exercise answer Exercise 4.7.14.)
20
Find solutions to \(3x-4\equiv 0\) (mod \(49\)) and (mod \(343\)) using the method in Subsection 6.5.2, starting with modulus seven.
21
Fill in the details of Example 6.5.2.
In the next few exercises, recall the definition of least common multiple (or lcm) from Exercise 2.5.9.
22
Find the pairwise least common multiples (recall Exercise 2.5.9) in the previous few exercises.
23
Find a formula for the lcm using the FTA: \begin{equation*}\text{lcm}(a,b)=\prod_{i=1}^n p_i^{???}\end{equation*}
24
Prove that if \(a,b>0\) then \(\gcd(a,b)\text{lcm}(a,b)=ab\) using the FTA.
25
Let \(\mathbb{Z}[\sqrt{-5}]\) be the set of all numbers of the form \(a+b\sqrt{-5}\) for \(a,b\in\mathbb{Z}\). Find two different factorizations of \(N=6\) in this set (known as a ring), , all the factors of which are not \(\pm 1\).