How many 2-digit positive integers are there such that the product

How many $2$-digit positive integers are there such that the product of their two digits is $24$ ?

  1. One
  2. Two
  3. Four
  4. Six
  5. Eight

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 9 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of Counting Methods can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

We want to count the number of two-digit positive integers satisfying a specific criterion, so we’ll likely draw on what we’ve learned about Counting Methods in math. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

 

  1. We want to know the number of $2$-digit positive integers for which the product of their digits is $24$

 

Develop a Plan

Well, seems like a straightforward question. First, let’s make a list of the positive factors of $24$, since it isn’t a very long list. We know that factors of a number $x$ are the numbers that can evenly divide into $x$. Or put another way, factors of a number $x$ are numbers where if we divide $x$ by that number, we’ll get an integer as a result. For example, $2$ is a factor of $8$, since $8/2=4$, and $4$ is an integer. But $2$ is NOT a factor of $9$, since $9/2=4.5$, and $4.5$ is NOT an integer. So for positive factors of $24$, we know that:

$24$ $=$ $1·24$
$ $ $ $
$24$ $=$ $2·12$
$ $ $ $
$24$ $=$ $3·8$
$ $ $ $
$24$ $=$ $4·6$
$ $ $ $

So the factors of $24$ are: $1, 2, 3, 4, 6, 8, 12, \and 24$. Next, we’re looking for two-digit positive numbers where the digits multiply to $24$. So we only want to keep the factor pairs we found above that are both single-digits. This leaves us with $3$ paired with $8$, and also $4$ paired with $6$. Let’s figure out how to combine these to get a complete list of the two-digit positive integers satisfying the constraint in this question.

Solve the Question

One trick we need to realize is that for any two-digit integer we get here, we can reverse the order of the integers. After all,
if $3·8=24$, then sure enough $8·3=24$ too. So for the two pairs of integers ($3$ and $8$, along with $4$ and $6$), the two-digit positive integers whose product of their digits is $24$ are: $38, 83, 46, \and 64$. Since we have four different two-digit integers, the correct answer is C, Four.

What Did We Learn

Always have to be careful with Counting Methods questions to make sure we don’t skip any possibilities. Looks like the trap being laid for us in this question was forgetting to reverse the order of the integers. Easy to see how someone might come up with $38$ and $46$ as the two-digit integers and mistakenly choose “Two” as the right answer. Being careful and actively looking for all possibilities are great tips for avoiding this kind of mistake.

 

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