Abstract Algebra: Theory and Applications

Prove that $n!>2^n$ for $n\ge 4\text{.}$

Prove that ${10}^{n+1}+{10}^n+1$ is divisible by $3$ for $n\in \mathbb{N}\text{.}$

Prove that $4\cdot {10}^{2n}+9\cdot {10}^{2n-1}+5$ is divisible by $99$ for $n\in \mathbb{N}\text{.}$

Use induction to prove that $1+2+2^2+\cdots +2^n=2^{n+1}-1$ for $n\in \mathbb{N}\text{.}$

If $x$ is a nonnegative real number, then show that $(1+x{)}^n-1\ge nx$ for $n=0,1,2,\dots \text{.}$

Prove that the two principles of mathematical induction stated in Section 2.1 are equivalent.

Show that the Principle of Well-Ordering for the natural numbers implies that 1 is the smallest natural number. Use this result to show that the Principle of Well-Ordering implies the Principle of Mathematical Induction; that is, show that if $S\subset \mathbb{N}$ such that $1\in S$ and $n+1\in S$ whenever $n\in S\text{,}$ then $S=\mathbb{N}\text{.}$

Let $a$ and $b$ be nonzero integers. If there exist integers $r$ and $s$ such that $ar+bs=1\text{,}$ show that $a$ and $b$ are relatively prime.

Let $x,y\in \mathbb{N}$ be relatively prime. If $xy$ is a perfect square, prove that $x$ and $y$ must both be perfect squares.

Using the division algorithm, show that every perfect square is of the form $4k$ or $4k+1$ for some nonnegative integer $k\text{.}$

Let $n\in \mathbb{N}\text{.}$ Use the division algorithm to prove that every integer is congruent mod $n$ to precisely one of the integers $0,1,\dots ,n-1\text{.}$ Conclude that if $r$ is an integer, then there is exactly one $s$ in $\mathbb{Z}$ such that $0\le s<n$ and $[r]=[s]\text{.}$ Hence, the integers are indeed partitioned by congruence mod $n\text{.}$

Define the least common multiple of two nonzero integers $a$ and $b\text{,}$ denoted by $\mathrm{lcm}(a,b)\text{,}$ to be the nonnegative integer $m$ such that both $a$ and $b$ divide $m\text{,}$ and if $a$ and $b$ divide any other integer $n\text{,}$ then $m$ also divides $n\text{.}$ Prove there exists a unique least common multiple for any two integers $a$ and $b\text{.}$

If $d=gcd(a,b)$ and $m=\mathrm{lcm}(a,b)\text{,}$ prove that $dm=|ab|\text{.}$

Show that $\mathrm{lcm}(a,b)=ab$ if and only if $gcd(a,b)=1\text{.}$

Prove that $gcd(a,c)=gcd(b,c)=1$ if and only if $gcd(ab,c)=1$ for integers $a\text{,}$ $b\text{,}$ and $c\text{.}$

Let $a,b,c\in \mathbb{Z}\text{.}$ Prove that if $gcd(a,b)=1$ and $a\mid bc\text{,}$ then $a\mid c\text{.}$

Let $p\ge 2\text{.}$ Prove that if $2^p-1$ is prime, then $p$ must also be prime.

Prove that there are an infinite number of primes of the form $6n+5\text{.}$

Prove that there are an infinite number of primes of the form $4n-1\text{.}$

Using the fact that $2$ is prime, show that there do not exist integers $p$ and $q$ such that $p^2=2q^2\text{.}$ Demonstrate that therefore $\sqrt{2}$ cannot be a rational number.