# How to prove that there is no positive integer solution to this two variables inequalities system? [closed]

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.

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

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.

• Where can I learn about the theorem or proof that you gave me, Rajan? Feb 12, 2020 at 9:06
• 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. Feb 12, 2020 at 9:19

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 $$(2)$$.

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