Articles of elementary number theory

Prove by contradiction that $\forall x,y \in \Bbb Z: x^2-4y \ne 2$

Prove that for all $x,y \in \mathbb{Z}$, $x^2 – 4y \ne 2$. Using a contradictory method would be appropriate. So, for this question, I assume, for the sake of a contradiction, that There exists $x,y \in \mathbb{Z}$ such that $x^2 – 4y = 2$. After this, I have to derive a contradiction somehow. I’m not […]

From any ten naturals, find some numbers whose sum is divisible by $ 10.$

Consider $A \subset \mathbb N $ such that $|A| = 10.$ Then prove that there exists a non-empty $B \subseteq A$ such that the sum of the elements in $B$ is divisible by $10.$ How to go to the gist of this question? Thanks in advance…

$Ord_n(ab)$ when $(a,n)=(b,n)=1$ but $(Ord_n(a), Ord_n(b))\neq 1$

What can be said about $Ord_n(ab)$ when $a,b$ are positive integers both relatively prime to $n$ and $Ord_n(a)$ is not relatively prime to $Ord_n(b)$? To start the proof I let $r=Ord_n(a)$, $s=Ord_n(b)$, and $t=Ord_n(ab)$. Since the orders of $a$ and $b$ are not relatively prime, I let $(r,s)=d$. Then I calculated the following: $$(ab)^{rs/(r,s)}\equiv a^{rs/d}b^{rs/d} […]

Show that $a^{61} \equiv a\ (mod\ 1001)$ for every $a \in \mathbb{N}$

I’m asked to show that $a^{61} \equiv a\ (mod\ 1001)$ for every $a \in \mathbb{N}$. I’ve tried to tackle this using Fermat’s Little Theorem and Euler’s Theorem, but I can’t even get started. My main problem seems to be the “for every $a \in \mathbb{N}$” part, because if it restricted $gcd(a, 1001) = 1$ I […]

Quarter circle train tracks

A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. […]

Why does $\gcd(a, b) = \min\lbrace ma + nb : m, n\in\mathbb{Z}\text{ and }ma+nb>0\rbrace$?

$\gcd(a, b)$ should have the form of $ma+nb$, where $m,n\in\mathbb{Z}$, since $(a, b)$ divides both $a$ and $b$. But I dont know why it should be the smallest one which is positive.

The remainder when $33333\ldots$ ($33$ times) is divided by $19$

$A= 33333\ldots$ ($33$ times). What is the remainder when $A$ is divided by $19$? I don’t know the divisibility rule of $19.$ What I did was $32\times(33333\times100000)/19$ and my remainder is not zero and this is completely divisible by $19.$

Showing $x^3-b$ is ireducible over $F_k$ given some conditions

I am self learning some number theory, and came upon this question which i cannot solve. Can someone give me a solution to thie, along with an explanation? I have been going around in circles and coming up with nothing usefull. Let k be a prime power and let $b$ be an element of $(F_k)$*. […]

$p$-adics, elements of $\mathbb{Q}_5^\times/(\mathbb{Q}_5^\times)^2$?

Here is a question surrounding the $p$-adics. I am curious as to what the description of the quotient group $\mathbb{Q}_5^\times/(\mathbb{Q}_5^\times)^2$ is, i.e. what are its elements? Here is an idea. I know that there is an isomorphism of groups $\mathbb{Q}_p^\times/\mathbb{Z}_p^\times = \mathbb{Z}$ which sends the class of $p^n$ ($n \in \mathbb{Z}$) to $n$, if that […]

Prove that it is impossible to have integers $b=5a$ under a digit re-ordering

Let $a$ be a positive integer and let $b$ be obtained from a by moving the initial digit of $a$ to the end. Prove that it is impossible to have $b=5a$.