TuesdACT Video: Red Book Math Test 4 #48 – Integers


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

F. 3^n
G. n^3
H. 3n
J. n/3
K. 3 + n

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 n. Don’t forget that we don’t know whether n is even or odd, so we need to test for both cases.

Answer Choice F:

3 is raised to a power of n. If n is even (2), this gives us 3 x 3 (ODD x ODD = ODD). If n 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:

n is raised to a power of 3. So if n 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 n is even then we have 2 x 2 x 2, and EVEN x EVEN = EVEN. So n^3 is not necessarily odd.

Answer Choice H:

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

Answer Choice J:

n/3. This is an interesting one, because if we pick a value for n 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:

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

So our answer is F.


  • Kristin Fracchia

    Dr. Kristin Fracchia has over fifteen years of expertise in college and graduate school admissions and with a variety of standardized tests, including the ACT, SAT, GRE, GMAT, and LSAT, with several 99% scores. She had a PhD from the University of California, Irvine, an MA degree from The Catholic University, and BA degrees in Secondary Education and English Literature from the University of Maryland, College Park. She was the recipient of the 2013 Excellence in Teaching Award and the Chancellor’s Club Fellowship from the University of California, Irvine. She’s worked as a high school teacher and university professor, as an independent college and graduate school admissions counselor, and as an expert tutor for standardized tests, helping hundreds of students gain acceptance into premier national and international institutions. She now develops accessible and effective edtech products for Magoosh. Her free online content and YouTube videos providing test prep and college admissions advice have received over 6 million views in over 125 countries. Kristin is an advocate for improving access to education: you can check out her TEDx talk on the topic. Follow Kristin on LinkedIn!

By the way, Magoosh can help you study for both the SAT and ACT exams. Click here to learn more!

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