6.E: Prime Numbers (Exercises)

Last updated
Save as PDF

Page ID: 7605

This page is a draft and is under active development.

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Exercise \(\PageIndex{1}\):

Are 253 and 257 prime?

Exercise \(\PageIndex{2}\):

Using prime factorization find the GCD and LCM of 3920 and 820.

Exercise \(\PageIndex{3}\):

Using prime factorization find the GCD and LCM of \(30030\) and \(165\).

Exercise \(\PageIndex{4}\):

Find the prime factorization of \(10101\).

Answer: \(3 \times 7 \times 13 \times 37 \).

Exercise \(\PageIndex{5}\):

1. Let \(a\) and \(b\) be positive integers such that \(a^2|b^2\). Show that \(a|b\).

2. Let \(a\) and \(b\) be positive integers. Prove that \(\gcd (a^2,b^2)=(\gcd(a,b)^2\).

3. Let \(a\) and \(b\) be positive integers such that \(\gcd(a,b)=1\). If \(ab\) is a perfect square then show that \(a\) and \(b\) are both perfect square.

Hint: Use the fundamental theorem of Arithmetic.

Exercise \(\PageIndex{6}\):

Describe in terms of the prime numbers all numbers with exactly four divisors.

Exercise \(\PageIndex{7}\):

1. Find a prime \(k\) such that \( 2^k-1\) is not a prime.

2. Find an integer \(k\), which is a power of 2 such that \( 2^k+1\) is not a prime.

Answer: 1. k=29, 2. \(k=32\)