Use a direct proof to show that the product of two odd integers is odd.

Step-by-step explanation:A direct proof is a method that takes an statement p, which we assume to be true, and use it to show directly that another statement q is true. So this method has the following steps:Assume the statement p is trueUse what we know about p and other facts as necessary to deduce that another statement q is true, that is show p ⇒ q is true.Fact that we need to use:Every odd integer can be written in the form 2m + 1 for some unique other integer mLet p be the statement a and b be odd integers and q be the statement that the product of a and b is odd.Proposition if a and b are odd, then the product of a and b is oddProof: Assume that a and b are odd integers, the by definition a = 2m + 1 and b = 2n + 1 for some integers m and n. we will now use this to show that the product of a and b is odd.[tex]a\cdot b= (2m+1) \cdot (2n+1)\\a\cdot b = 2m\cdot 2n+2m+2n+1\\a\cdot b =4mn+2m+2n+1\\a\cdot b = 2(2mn+2m+2n) +1\\\:If  \:k=2mn+2m+2n\\a\cdot b = 2k+1[/tex] Hence we have shown that the product of a and b is odd since 2k + 1 is and odd integer. Therefore we have shown that p ⇒ q and so we have completed our proof.