Linked Questions

89
votes
5answers
22k views

Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
66
votes
6answers
10k views

Why is $9$ special in testing divisiblity by $9$ by summing decimal digits? (casting out nines)

I don't know if this is a well know fact but I have observed that every number no matter how large that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there ...
31
votes
13answers
18k views

How to avoid arithmetic mistakes?

When dealing with several numbers and long equations, it's common to make careless arithmetic mistakes that give the wrong answer. I was wondering if anyone had tips to catch these mistakes, or even ...
38
votes
7answers
21k views

Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is $\...
33
votes
4answers
37k views

Can decimal numbers be considered “even” or “odd”?

Is the concept of even/odd numbers is applicable to decimal numbers? For e.g. - 4.222 is a even number?
30
votes
8answers
14k views

Why $\gcd(qb+r,b)=\gcd(b,r)$?

Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
15
votes
4answers
18k views

Can you construct a field with 4 elements?

Can you construct a field with 4 elements? can you help me think of any examples?
7
votes
10answers
14k views

Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd

If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots? I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and rational numbers are of the form $\...
11
votes
4answers
3k views

Proof by contradiction: $r - \frac{1}{r} =5\Longrightarrow r$ is irrational?

Prove that any positive real number $r$ satisfying: $r - \frac{1}{r} = 5$ must be irrational. Using the contradiction that the equation must be rational, we set $r= a/b$, where a,b are positive ...
12
votes
4answers
3k views

Are these two quotient rings of $\Bbb Z[x]$ isomorphic?

Are the rings $\mathbb{Z}[x]/(x^2+7)$ and $\mathbb{Z}[x]/(2x^2+7)$ isomorphic? Attempted Solution: My guess is that they are not isomorphic. I am having trouble demonstrating this. Any hints, as to ...
5
votes
5answers
11k views

If the square of a number is even, then the number if even. Isn't that not true for 2?

I'll quickly go over my understanding of it: If a number $n^2$ is even, then $n$ is even. The contrapositive is that is that if $n$ is not even (odd), then $n^2$ must also be not be even (be odd). ...
3
votes
7answers
2k views

Test for an Integer Solution

This came up an a training piece for the Putnam Competition and also in Ireland and Rosen. The question posed was basically: Let $p(x)$ be a polynomial with integer coefficients satisfying that $p(0)...
9
votes
3answers
572 views

How are the integral parts of $(9 + 4\sqrt{5})^n$ and $(9 − 4\sqrt{5})^n$ related to the parity of $n$?

I am stuck on this question, The integral parts of $(9 + 4\sqrt{5})^n$ and $(9 − 4\sqrt{5})^n$ are: even and zero if $n$ is even; odd and zero if $n$ is even; even and one if $n$ is ...
3
votes
6answers
3k views

Polynomials with integer coefficients

Through definitions, theorems and my professor the following is true: The product of any two odd integers is odd. The sum and difference of any two odd integers are even. The sum, product and ...
4
votes
5answers
267 views

Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field

Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field. How can one justify the answer in the shortest number of lines?

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