How to prove that there is no positive integer solution to this two variables inequalities system? { 77 n − 154 m ≤ − 1 154 m − 77 n ≤ 76

Question

How to prove that there is no positive integer solution to this two variables inequalities system?      {                            77          n                          −          154          m                          ≤          −          1                                      154          m                          −          77          n                          ≤          76

Cristian Hamilton · Accepted Answer

The given inequalities can be written as  1  ≤  154  m  −  77  n  ≤  76      ,  w  h  e  r  e    m  ,  n  ∈  N  0  &amp;lt;      1    77    ≤  2  m  −  n  ≤      76    77    &amp;lt;  1  ∀  m  ,  n  ∈  N      ,  2  m  −  n  ∈  IFor any positive integer   m  ,  n      ,  2  m  −  n  ∉  (  0  ,  1  )Hence therefore there doesnot exist any positive integer   m  ,  n that satisfy both inequality.

Yahir Tucker · Accepted Answer

The first inequality gives   n  −  2  m  ≤  −      1    77    . Since   n  −  2  m is an integer, we get(1)   (  1  )    n  −  2  m  ≤  −  1.The second inequality gives   2  m  −  n  ≤      76    77    . Since   2  m  −  n is an integer, we get(2)   (  2  )    2  m  −  n  ≤  0..But from (1) it follows that   2  m  −  n  ≥  1.. This is a contradiction to (2).

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

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