At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

If [tex]$9p + 5$[/tex] is even, which of the following must be even?

A. [tex]$p + 5$[/tex]
B. [tex][tex]$2p + 5$[/tex][/tex]
C. [tex]$5p$[/tex]
D. [tex]$9p$[/tex]


Sagot :

To determine which of the given expressions must be even if [tex]\(9p + 5\)[/tex] is even, let's analyze the given condition step by step.

First, understand that for [tex]\(9p + 5\)[/tex] to be even, [tex]\(9p\)[/tex] itself must satisfy certain conditions. Here's how we can reason through the problem:

1. Given condition: [tex]\(9p + 5\)[/tex] is even.

2. If we subtract 5 from both sides of the given condition, we get:
[tex]\[ 9p + 5 - 5 = \text{even number} - 5 \implies 9p = \text{even number} - 5 \][/tex]

3. An even number minus an odd number results in an odd number. Thus:
[tex]\[ 9p = \text{odd number} \][/tex]

4. Since [tex]\(9p\)[/tex] is odd, [tex]\(p\)[/tex] itself must be odd. This is because 9 is an odd number, and for the product of two numbers to be odd, both numbers must be odd.

With [tex]\(p\)[/tex] confirmed to be odd, let's investigate each of the given options:

Option A: [tex]\(p + 5\)[/tex]

If [tex]\(p\)[/tex] is odd, adding 5 (which is also odd) to it will result in an even number since:
[tex]\[ \text{odd} + \text{odd} = \text{even} \][/tex]
Hence, [tex]\(p + 5\)[/tex] is even when [tex]\(p\)[/tex] is odd.

Option B: [tex]\(2p + 5\)[/tex]

If [tex]\(p\)[/tex] is odd, multiplying it by 2 will result in an even number since:
[tex]\[ 2 \times \text{odd} = \text{even} \][/tex]
However, adding 5 (an odd number) to an even number will result in an odd number since:
[tex]\[ \text{even} + \text{odd} = \text{odd} \][/tex]
Therefore, [tex]\(2p + 5\)[/tex] is odd when [tex]\(p\)[/tex] is odd.

Option C: [tex]\(5p\)[/tex]

If [tex]\(p\)[/tex] is odd, multiplying it by 5 (an odd number) will result in an odd number since:
[tex]\[ \text{odd} \times \text{odd} = \text{odd} \][/tex]
Hence, [tex]\(5p\)[/tex] is odd when [tex]\(p\)[/tex] is odd.

Option D: [tex]\(9p\)[/tex]

We already established that [tex]\(9p\)[/tex] is odd if [tex]\(p\)[/tex] is odd, since [tex]\(9\)[/tex] (an odd number) multiplied by another odd number results in an odd number:
[tex]\[ \text{odd} \times \text{odd} = \text{odd} \][/tex]
Hence, [tex]\(9p\)[/tex] is odd.

In summary, among the given options, the expression that must be even is:

[tex]\[ \boxed{p+5} \][/tex]