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$\mathrm{K}$ is an even number and $\mathrm{P}$ is an odd number. Which of the following statements is not correct?

(1) $\mathrm{P}-\mathrm{K}-1$ is an odd number.
(2) $\mathrm{P}+\mathrm{K}+1$ is an even number.
(3) $\mathrm{P} \cdot \mathrm{K}+\mathrm{P}$ is an odd number.
(4) $\mathrm{P}^{2}+\mathrm{K}^{2}+1$ is an even number.
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Let us examine each of the statements:

(1) The difference between an odd number (P) and an even number (K) is an odd number. Therefore, $\mathrm{P}-\mathrm{K}$ is an odd number. If we subtract 1 from this odd number, we get an even number. Therefore, $\mathrm{P}-\mathrm{K}-1$ is an even number, and the statement is not correct.

(2) The sum of an odd number (P) and an even number (K) is an odd number. Therefore, $\mathrm{P}+$ $\mathrm{K}$ is an odd number. If we add 1 to this odd number, we get an even number. Therefore, $\mathrm{P}$ $+\mathrm{K}+1$ is an even number, and the statement is correct.

(3) The product of an even number and any integer is always an even number; therefore, $\mathrm{P} \cdot \mathrm{K}$ is an even number. If we add an odd number to this even product, we get an odd number. Therefore, $\mathrm{P} \cdot \mathrm{K}+\mathrm{P}$ is an odd number and the statement is correct.

(4) The square of an odd number $\left(P^{2}\right)$ is an odd number because it is the product of an odd number multiplied by an odd number $(\mathrm{P} \cdot \mathrm{P})$. The square of an even number $\left(\mathrm{K}^{2}\right)$ is an even number because it is the product of an even number multiplied by an even number $(\mathrm{K} \cdot \mathrm{K})$. The sum of the two squared numbers $\left(\mathrm{P}^{2}+\mathrm{K}^{2}\right)$ is odd because it is the sum of an odd number and an even number. Therefore, if we add 1 to this sum, we get an even number. Thus, $\mathrm{P}^{2}+\mathrm{K}^{2}+1$ is an even number, and the statement is correct.

In this question, you are asked to choose the statement that is not correct; therefore, (1) is the correct response.
by Diamond (88,180 points)

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