-2
$\begingroup$

I would like to know if it is possible to prove that there are no integer solutions to: $$ \begin{cases} 77n &- 154m &\leq -1\\ 154m &- 77n &\leq 76 \end{cases} $$ I have no idea how to start, so any comments are welcome. Thank you and regards, Tony.

$\endgroup$
4
  • 2
    $\begingroup$ Use that $154$ is divisible by $77$. $\endgroup$ Feb 12, 2020 at 8:05
  • $\begingroup$ yes, i need to prove that there are no integer solutions for n and m. $\endgroup$
    – gu0hu1
    Feb 12, 2020 at 8:10
  • $\begingroup$ What is a number of natural numbers on $[1,76]$, which are divisible by $77$? $\endgroup$ Feb 12, 2020 at 8:12
  • $\begingroup$ how about 154y - 154x <= -1 and 154m - 154m <=76? how to find x and integers? $\endgroup$
    – gu0hu1
    Feb 12, 2020 at 8:19

2 Answers 2

2
$\begingroup$

The given inequalities can be written as $$1\le 154m-77n\le 76\;\;,where\,m,n\in N$$ $$0\lt\frac{1}{77}\le 2m-n\le \frac{76}{77}\lt 1$$ $$\forall m,n \in N \;\;,2m-n\in I $$ For any positive integer $m,n\;\;, 2m-n\notin (0,1)$ $$. $$ Hence therefore there doesnot exist any positive integer $m,n$ that satisfy both inequality.

$\endgroup$
2
  • $\begingroup$ Where can I learn about the theorem or proof that you gave me, Rajan? $\endgroup$
    – gu0hu1
    Feb 12, 2020 at 9:06
  • $\begingroup$ These all are part of school level mathematics for which one more site which I follow is. mathsdiscussion.com. You can find some and also learn there. $\endgroup$
    – Rajan
    Feb 12, 2020 at 9:19
1
$\begingroup$

The first inequality gives $n-2m \le -\frac{1}{77}.$ Since $n-2m$ is an integer, we get

$(1) \quad n-2m \le -1.$

The second inequality gives $2m-n \le \frac{76}{77}.$ Since $2m-n$ is an integer, we get

$(2) \quad 2m-n \le 0.$

But from $(1)$ it follows that $2m-n \ge 1.$ This is a contradiction to $(2)$.

$\endgroup$
1
  • $\begingroup$ Thank you so much, Fred. Are there any references for the theorem that I can read to prove the contradiction? $\endgroup$
    – gu0hu1
    Feb 12, 2020 at 8:31

Not the answer you're looking for? Browse other questions tagged .