If m and p are positive integers and (m + p)x m is even, which of the following must be true? Would it be: (A) if m is odd, then p is odd. (B) if m is odd, then p is even. (C) if m is even, then p is even and (D) if m is even, then p is odd. I had chosen D and that was wrong. Would it be C?

for the product to be even, you must have either

m even
or
m+p even

A works, because odd+odd is even
B fails -- odd*odd = odd
C works whether p is even or odd, so it's not true that p must be even.
D the same

Ah, the wonderful world of integers! Alright, let's analyze this situation using a touch of humor.

If (m + p)x m is even, then either (m + p) or m must be even. Why, you ask? Well, because when an odd number is multiplied by any other number, the result is always odd. So, if (m + p)x m is even, at least one of those factors better be even, or else we'd be dealing with some magical math here!

Now, let's take a closer look at the options. We've got:

(A) If m is odd, then p is odd.
(B) If m is odd, then p is even.
(C) If m is even, then p is even.
(D) If m is even, then p is odd.

Since we know that at least one of the factors needs to be even, (C) is actually the correct answer! If m is even, then it guarantees that (m + p)x m is even, and since both m and p must be even, it satisfies the condition.

So, congratulations! Option (C) is the one that brings a smile to our math-loving faces. Keep up the great work!

To determine which of the statements is true, we can consider the various possibilities for the parity (even or odd) of m and p.

Let's first consider the case when m is odd:

In this case, we have (odd + p) × odd = even.

For the product to be even, p must be odd, because the sum of an odd number and an even number is odd. Therefore, statement (A) "if m is odd, then p is odd" is true.

Now, let's consider the case when m is even:

In this case, we have (even + p) × even = even.

Regardless of the parity of p, the sum of an even number and any other number is even. Therefore, both p can be even or odd.

So, statement (C) "if m is even, then p is even" is not necessarily true, because p can also be odd.

Therefore, the correct answer is (A) if m is odd, then p is odd.

To determine the correct answer, let's break down the given information:

We are given that (m + p) * m is even, where m and p are positive integers.

To begin, let's examine the properties of even numbers:
- An even number can be divided by 2 without leaving a remainder.

Now, let's look at the properties of odd numbers:
- An odd number cannot be divided by 2 without leaving a remainder.

Using these properties, we can analyze the options:

(A) If m is odd, then p is odd.
If m is odd, we have (odd + p) * odd. Adding an odd number to p will always result in an odd number. However, when we multiply an odd number by another odd number, the result is always an odd number, not an even number. Therefore, option (A) is not necessarily true and can be eliminated.

(B) If m is odd, then p is even.
Using the same reasoning, if m is odd, we have (odd + p) * odd. Adding an odd number to p will always result in an odd number. Multiplying an odd number by an odd number results in an odd number. This means that the product cannot be even. Therefore, option (B) is not necessarily true and can be eliminated.

(C) If m is even, then p is even.
If m is even, (even + p) * even could result in an even number. Adding an even number to p can result in either an even or odd number, depending on p. However, when multiplying an even number by another even number, the product is always an even number. Thus, if m is even, it is possible for the product to be even. Therefore, option (C) is potentially true and should be considered.

(D) If m is even, then p is odd.
Testing this statement by assuming m is even, we have (even + p) * even. Adding an even number to p would not guarantee an odd result. It could be either even or odd. Multiplying an even number by an even number always results in an even number. Therefore, option (D) is not necessarily true and can be eliminated.

Based on our analysis, the correct answer would be option (C): If m is even, then p is even.