Quantity A |
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | Quantity B |

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | ||

The greatest prime factor of $1{,}000$ | $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ | The greatest prime factor of $68$ |

- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.

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 4 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of **Integers** 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 compare the greatest **prime factors** between the two quantities, which tells us that it likely tests our **Integers** math skill. 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.

- We need to find the greatest prime factor of $1{,}000$
- We need to find the greatest prime factor of $68$
- We want to compare these values

## Develop a Plan

We want the greatest prime factors for these numbers. The factors of $1{,}000$ are numbers that multiply together to give us $1{,}000$. Since we’ll be looking for the *greatest* prime factor, let’s ignore any negative factors of our numbers. Let’s quickly review prime numbers since it’s crucial to this question. Of course, if we’re comfortable with prime numbers, we can just skip this refresher.

## Concept Refresher – Prime Numbers

A **prime number** is any number that is only divisible by $1$ and itself. If an integer is not a prime number, then we call it a **composite number**. “Divisible” means that when dividing by a certain number, there is no remainder. For example,

$12$ is evenly divisible by $3$ because $3$ goes into $12$ four times with no remainder. However, $14$ is not evenly divisible by $3$, because $3$ goes into $14$ four times but has a remainder of $2$.

$2$ is the first prime number. It also holds the honor of being the only even prime number. The next few prime numbers are: $3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41$. The list can go on and on and on. So what’s the best way to recognize prime numbers? The two best tips for recognizing prime numbers are: 1) Learning the $9·9$ multiplication table and 2) Learning the divisibility rules for $2, 3, \and 5$ (remembering that if a number is divisible by anything other than $1$ and itself, it’s NOT a prime number.

By learning the $9·9$ multiplication table well, we will immediately recognize many numbers as NOT being prime numbers. For example, if we know that $63=9·7$, then we’ll immediately recognize $63$ as NOT being a prime number.

A number is divisible by $2$ if it is even. That is to say, if its units digit (the one to the left of the decimal point) is $0, 2, 4, 6, \or 8$. So for example, $16$ is divisible by $2$ because its units digit is $6$, but $17$ is NOT divisible by $2$

A number is divisible by $3$ if the sum of its digits is divisible by $3$. For example, $57$ is divisible by $3$ because $5+7=12$, and $12$ is divisible by $3$. This rule will help us recognize many fairly large numbers that are NOT prime numbers, but also do not appear on our $9·9$ multiplication table.

A number is divisible by $5$ if its units digit is either $0 \or 5$. So for example, $65$ is divisible by $5$, but $66$ is NOT divisible by $5$. Now that we’ve reviewed prime numbers, let’s get back to the question at hand!

So our plan will be to find the prime factors of both quantities. We can do this fairly quickly by thinking of numbers that multiply together to give us the values in these quantities, and step-by-step we can break those down into smaller numbers until we end up with prime numbers.

## Solve the Question

First let’s do the prime factorization for $1{,}000$. We know that when a number ends in $0$, it is divisible by $10$. So let’s factor out three $10\s$.

$$1{,}000 = 10·10·10$$

We know that $10=2·5$, so let’s complete the prime factorization using this.

$$1{,}000 = 5·2·5·2·5·2$$

Prime factorization complete! Okay, so for $1{,}000$ it looks like the greatest prime factor is $5$. Now let’s do the prime factorization for $68$. We know that $68$ is even, so it must be divisible by $2$. Dividing $68$ by $2$, feeling free to use our calculator if necessary, we get $68/2 = 34$. So let’s split $68$ into these two factors.

$$68 = 2·34$$

Since $34$ is *also* even, let’s divide it by $2$ to get $34/2=17$. Let’s continue factoring.

$$68 = 2·2·17$$

We can’t factor $17$ any further, so it is a prime factor. So the greatest prime factor of $68$ is $17$.

Since the greatest prime factor for Quantity A $(5)$ is smaller than the greatest prime factor for Quantity B $(17)$, **the correct answer is B, Quantity B is greater**.

## What Did We Learn

We learned prime factorization! This is a very useful skill to master for GRE questions testing the **Integers** math skill. We can improve our prime factoring ability by: 1) memorizing the $9·9$ multiplication table and 2) memorizing the divisibility rules for $2, 3, \and 5$ that we reviewed earlier.

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