Chapter 1: Arithmetic in Z Revisited

If p is a positive prime, prove that$\sqrt{p}$ is irrational.

(Euclid) Prove that there are infinitely many primes.

_{If $a>o$and $b>0$, prove that $\left[a,b\right]=\frac{ab}{\left(a,b\right)}$.}

Let $p>1$ If $2^p-1$ is prime, prove that pis prime.

Prove or disprove: If n is an integer and $n>2$ , then there exists a prime p such that role="math" localid="1646249529428" $n<p<n!$.

1. Let a be a positive integer. If $\sqrt{a}$ is rational, prove that $\sqrt{a}$ is an integer.
2. Let rbe a rational number and a an integer such that $r^n=a$ . Prove that r is an integer. [Part (a) is the case when$n=2$ ].

Letp, q be primes with$p\ge 5,q\ge 5$ . Prove that$24|\left(p^2-q^2\right)$ .

In Exercises 3 and 4, use a calculator to find the quotient q and remainder r when a is divided by b.

(a) $a=8,126,493;b=541$

(b) $a=-9,217,645;b=617$

(c)$a=171,819,920;b=4321$

(a) If $a|b$ and $a|c$, prove that$a|\left(b+c\right)$.

(b) If $a|b$ and $a|c$, prove that$a|\left(br+ct\right)$for any $r,t\in \mathbb{z}$.

Let a be any integer and let b and e be positive integers. Suppose that when a is divided by b, the quotient is q and the remainder is r, so that

$a=bq+r$ and$0\le r<b$.

If ae is divided by bc, show that the quotient is q and the remainder is rc.