The system of inequalities is as follows: { a − ⌊ a b ⌋ ⋅ b ≤ M a − ⌊ a b ⌋ ⋅ b ≥ m I need to solve for a and b. M and m are known constants.

Question

The system of inequalities is as follows:      {                            a          −                      ⌊                                    a              b                        ⌋                    ⋅          b          ≤          M                                      a          −                      ⌊                                    a              b                        ⌋                    ⋅          b          ≥          m                        I need to solve for   a and   b.   M and   m are known constants.

Zayden Andrade · Accepted Answer

Hint: Write   a  =  b  q  +  r with   0  ≤  r  &amp;lt;  bLet   S be the set of all the solutions   (  a  ,  b  ).For the sake of simplicity let us seek the solutions such that   b  &amp;gt;  0. For any   a  ∈  R, one can write   a  =  b  q  +  r in a unique way, with   q  ∈  Z and   0  ≤  r  &amp;lt;  b. Obviously, the pair   (  a  ,  b  ) is a solution if and only if   m  ≤  r  ≤  M. Hence  {  (  a  ,  b  )  ∈  S  :  b  &amp;gt;  0  }  =  {  (  b  q  +  r  ,  b  )  :  max  (  0  ,  m  )  &amp;lt;  b   and   q  ∈  Z   and   r  ∈  [  0  ,  b  )  ∩  [  m  ,  M  ]  }of course, there are conditions to ensure that the interval   [  0  ,  b  )  ∩  [  m  ,  M  ] is non empty, namely   m  ≤  M and   m  &amp;lt;  b and   0  ≤  M.To investigate the case   b  &amp;lt;  0, define       a    ′    =  −  a and       b    ′    =  −  b

The system of inequalities is as follows: { <mtable columnalign="left left" rowspaci

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