Proving the Oddness of (y^{23y5}) When (y) is Odd: A Contrapositive Approach

Given that (y) is an integer and an odd number, we aim to prove that (y^{23y5}) is always odd. This proof will utilize a contrapositive approach, which inverses the statement and proves its validity.

Explanation of Odd and Even Numbers

To understand the proof, it is crucial to recognize the properties of odd and even numbers:

An even number can be written in the form (2n), where (n) is an integer. An odd number can be written in the form (2n 1), where (n) is an integer.

Contrapositive Proof Strategy

The contrapositive approach states that a statement (P) implies (Q) is true if its contrapositive, ( eg Q) implies ( eg P), is also true. In our case, we want to prove that if (y) is odd, then (y^{23y5}) is odd. The contrapositive would be to show that if (y^{23y5}) is even, then (y) must be even.

Step-by-Step Proof

Let's start by assuming (y) is odd and write it as (y 2n 1), where (n) is an integer.

Step 1: Substituting (y 2n 1) into the Expression

Substitute (y 2n 1) into the expression (y^{23y5}):

y  2n   1
y^{23y5}  (2n   1)^{2(2n   1)3(2n   1)5}

Simplify the exponent:

(23y5 2(2n 1) 3(2n 1) 5 6n 2 6n 3 5 12n 10)

y^{23y5}  (2n   1)^{12n   10}

Step 2: Proving the Oddness of (y^{12n 10})

Since (y 2n 1) is odd, raising an odd number to any integer power will result in an odd number. Hence, ((2n 1)^{12n 10}) is odd.

Step 3: Simplifying the Expression

Let's simplify the expression step by step to illustrate the odd nature:

((2n 1)^{12n 10} (2n 1)(2n 1)^{12n 9})

Since ((2n 1)^{12n 9}) is odd (as it is the product of odd numbers), multiplying it by another odd number ((2n 1)) results in an odd number.

Conclusion

Therefore, if (y) is odd, (y^{23y5}) is odd. This confirms the initial statement using the contrapositive approach. The contrapositive approach successfully proves that if (y^{23y5}) were even, (y) would need to be even, but we have established that if (y) is odd, (y^{23y5}) remains odd.

Additional Insight

It is also worth noting that if (y) were even, say (y 2m) for some integer (m), then the expression (y^{23y5}) would result in an even number because even raised to any positive integer power is always even. This ensures that the result depends on whether (y) is odd or even.

In summary, the proof of the oddness of (y^{23y5}) when (y) is odd is a clear demonstration of the properties of odd and even numbers and the power of using the contrapositive method in mathematical proofs.