Consider the claim: if m is an even integer, then m^2+5m-1 is an odd integer.
(a.) prove the claim using a direct proof.
(b.) state the converse. is the converse true? provide a proof or give a counterexample.
i have read the previous examples and am having trouble with them so don't copy and paste a retired answer.

Rewriting the expression using m=2p we have:

is an odd integer but the converse is not true.

Step-by-step explanation:

Even numbers are written as 2n where n is any integer, while odd numbers are written as 2n-1 where n is any integer.

a) To prove that is an odd integer, we have to prove that it can be written as 2n-1.

By hypothesis, m is an even integer so we will write it as 2p.

Rewriting the original expression using we have:

Solving the expression and factorizing it we get

And this last expression is an expression of the form 2n-1, and therefore is an odd integer.

b) The converse would be: if is an odd integer, then m is an even integer.

We'll give a counterexample, let's make , then

is an odd integer but m is odd.

Therefore, the converse is not true.

what are the options? ?

step-by-step explanation:

the 3rd option is right

step-by-step explanation: