Respuesta :

It is true that equation xy + x + y = 36 has no positive integers .

These are both forms of direct proof. There are many other types of proof. Examples of commonly used proof types include

  • Proof by contradiction: We assume that our conclusion is false, and then show that that must be false because it leads to a contradition.
  • Proof by contraposition: We use the fact that the truth of a proposition is equivalent to the truth of its contrapositive, and prove the contrapositive instead.
  • Proof by induction: We show that a proposition is true for some small integer, and show that its truth for one integer implies its truth for the next. We then claim that the proposition is true for all integers greater than our initial one using the property of mathematical induction.
  • Proof by cases: We divide up all possibilities into different cases, and then prove our claim for each one separately.
  • Proof by counterexample: We prove that a generalized claim is false by explicitly showing a case in which it is not true.

By using hit and trial method

So first we'll take which is the smallest positive integer .

put x = 1

⇒ y =(36-1)/2

   =   35/2  which is not an integer .

put x = 2

⇒ y = 34//3 which is again a fraction .

This proves equation xy + x + y = 36 has no positive integers .

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