2
votes
2answers
25 views

Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$

Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$ there was a hint which is use use contradiction.
3
votes
0answers
77 views

Prove the equality

Given $a,b,c,d$ are positive integers such that $a^2+b^2+c^2+d^2-ab-bc-cd-da$ is divisible by $abcd$. Prove that $a=b=c=d$.
2
votes
3answers
148 views

Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$

Let $a,b,c$ be positive integers. Verify that Diophantine equation $ax + by = c$ has integer solution $x_0, y_0$ if and only if $GCD(a,b)|c$. Attempt Diophantine $ax + by = c$ has integer solution ...
1
vote
2answers
59 views

Solving a problem with a diophantine equation without trial and error.

I have the following problem: A teacher bought toys for the students of an academy, every toy for a boys costs $290$ and every toy for a girl costs $330$. If he spends $24300$, how many of each ...
1
vote
1answer
62 views

Solve a system of diophantine equations

I have a problem with elemental number theory. I started with the expression $$ (a - \frac{1}{b})(b - \frac{1}{c})(c - \frac{1}{a}) $$ and task to find all natural $a,b,c$ so that the result of the ...
0
votes
1answer
45 views

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$?

What is the best way of solving for $x,y\in\mathbb N$ given the conditions $\begin{cases}x\mid y+a\\y\mid x + b\end{cases}$? The letters $a,b\in\mathbb N$ denote constant known numbers. The ...
1
vote
4answers
44 views

how do i prove $ab|n$ if $\gcd(a, b) = 1$ and $a|n $ and $b|n$?

Suppose that, for integers $a, b,$ and $n,$ $$\gcd(a, b) = 1\text{ and }a|n\text{ and }b|n.$$ How do I prove that $ab|n$ using linear Diophantine equations? Can I extend the above result to the ...
1
vote
1answer
81 views

Problem with Diophantine equation

Let $a,b \in \mathbb N$ be coprime. Prove that for all $n\in \mathbb N$ such that $n>ab$ there are $r,s\in \mathbb N$ such that $n=ra+sb$. I'm really stuck on this problem. I know that since ...
1
vote
3answers
96 views

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$.

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$. E.g. $p(7)=2\cdot3\cdot5$ (and $n=7$ is a solution). Let ...
4
votes
1answer
236 views

Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.

For what values of $m,n$ natural, do $2^n - 1$ is divisible by $2^{m+n} - 3^m$? Thank you very much.
2
votes
1answer
75 views

If $p$ is prime and $p>3$ and $k,l,m,n,p\in\mathbb N$ and $p^k+p^l+p^m=n^2$, prove that $8\mid p+1$.

If $k,l,m,n,p\in\mathbb N$ and a prime number $p>3$ that satisfies $$p^k+p^l+p^m=n^2$$ is chosen, prove that $8\mid p+1$. $n^2$, when divided by $8$, gives a remainder $1$ (it can't give the ...
1
vote
2answers
44 views

Is this a valid proof that $\{ax + by|x,y \in \Bbb Z\}= \{n\times \gcd(a, b) |n\in \Bbb Z\}$?

I'm trying to prove that $\{ax + by|x,y \in \Bbb Z\}= \{n\times \gcd(a, b) |n\in \Bbb Z\}$, but I'm unsure on the . The main proposition I'm using to solve this is that $\exists x,y, ax+by = \gcd(a, ...
1
vote
1answer
151 views

How to prove Greatest Common Divisor using Bézout's Lemma

The problem is to prove the following If $\gcd(a,b) = c$, then $\gcd(a^m, b^m) = c^m$ I know that this can be solved easily by proving that $c\mid a \implies c^m \mid a^m$ and $c\mid b \implies ...
0
votes
1answer
91 views

Find x when remainder is zero

Let $k$, $p$ and $r$ be constant values Find the $x$ with the smallest absolute value so that $$ \frac{k - r - x}{p + r + x}\in\mathbb{Z} $$
6
votes
3answers
417 views

The positive integer solutions for $2^a+3^b=5^c$

What are the positive integer solutions to the equation $$2^a + 3^b = 5^c$$ Of course $(1,\space 1, \space 1)$ is a solution.