# 6.E: Prime Numbers (Exercises)

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

## 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 $$10101$$.

$$3 \times 7 \times 13 \times 37$$.

## 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.

1. k=29, 2. $$k=32$$