In this episode of TuesdACT, we are taking a look at the principles of multiplying and adding even and odd numbers using an example from *The Real ACT Prep Guide* that lots of students struggle with.

*Real ACT Prep Guide* Test 4 Question 48 on page 599, we are coming for you!

Check out the video or read below for the explanation and a little lesson on even and odd integers.

If is a positive integer, which of the following expressions must be an odd integer?

F.

G.

H.

J.

K.

Alright, let’s pause and go over principles of adding, subtracting, and multiplying odd and even numbers. Here’s how it shakes out.

Even + Even = Even

Even + Odd = Odd

Odd + Odd = Even

Even x Even = Even

Even x Odd = Even

Odd x Odd = Odd

You don’t need to memorize the rules; if you forget, just go through the examples with easy numbers like 2 and 3 and see what happens. It will be the same for all even and odd numbers.

So, for example:

Even + Even = Even 2 + 2 = 4

Even + Odd = Odd 2 + 3 = 5

Odd + Odd = Even 3 + 3 = 6

Even x Even = Even 2 x 2 = 4

Even x Odd = Even 2 x 3 = 6

Odd x Odd = Odd 3 x 3 = 9

Question 48 is also a great example of how you can find important clues in the answer choices on the ACT. Notice that all of our answer choices have the number 3 in them, so all of our answer choices are doing something with an odd number.

Now let’s test with values for . Don’t forget that we don’t know whether is even or odd, so we need to test for both cases.

**Answer Choice F:**

3 is raised to a power of . If is even (2), this gives us 3 x 3 (ODD x ODD = ODD). If is odd (3), this gives us 3 x 3 x 3 (ODD x ODD x ODD), which is also always odd.

Since an odd number multiplied by an odd number is always an odd number, and because in this case we are just multiply 3 (an odd number) by itself a number of times, it will always be odd. So F is our answer, but let’s check the rest.

**Answer Choice G:**

is raised to a power of 3. So if is odd, then we have 3 x 3 x 3 (ODD x ODD x ODD) and we know that ODD x ODD = ODD. But if is even then we have 2 x 2 x 2, and EVEN x EVEN = EVEN. So is not necessarily odd.

**Answer Choice H:**

. So if is even (2), we have 3 x 2 (ODD x EVEN = EVEN); if is odd, we have 3 x 3 (ODD x ODD = ODD). So is not necessarily odd.

**Answer Choice J:**

. This is an interesting one, because if we pick a value for that does not divide evenly by 3, then it is actually __neither even nor odd__. By rule, the terms even or odd apply only to integers not to fractions.

**Answer Choice K:**

. So if is even (2), then we have 3 + 2 (ODD + EVEN = ODD); if is odd, then we have 3 + 3 (ODD + ODD = EVEN). So is not necessarily odd.

So our answer is F.

### More from Magoosh

##### About Kristin Fracchia

Dr. Kristin Fracchia currently focuses on our MCAT and LSAT Prep, but she also has expertise in a wide range of standardized tests, including the ACT, SAT, GRE, and GMAT, as well as college and grad school admissions. With a PhD from UC Irvine and degrees in Education and English, she’s been working in education since 2004. She enjoys the agony and bliss of long distance trail running, backpacking, hot yoga, and esoteric knowledge.

### Leave a Reply

**Magoosh blog comment policy:** To create the best experience for our readers, we will approve and respond to comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! :) If your comment was not approved, it likely did not adhere to these guidelines. If you are a Premium Magoosh student and would like more personalized service, you can use the Help tab on the Magoosh dashboard. Thanks!