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 12 of Section 6 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 your paper.
This question asks about prime numbers, which are a special type of integer, so this question 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 want to find the product $ab$, so we need $a$ and $b$
- $a$ is the smallest prime number greater than $21$
- $b$ is the largest prime number less than $16$
Develop a Plan
To get $ab$, we need $a$ and $b$. So let’s start by finding $a$. Since $a$ is the smallest prime number greater than $21$, we can start at $22$ and keep going up one integer at a time until we find a prime number. Similarly to find $b$, since it is the largest prime number less than $16$, we can start at $15$ and keep going down one integer at a time until we find a prime number. Then we can multiply $a$ and $b$ together to get $ab$. To determine if a number is prime, we’ll first check to see if it is on our $9×9$ multiplication table that we’ve memorized. If not, we’ll use divisibility rules for $2, 3, \and 5$ to further check if it is not a prime number. Let’s also quickly review prime numbers before we tackle this question. Of course, we can skip this refresher if we’re comfortable with prime numbers.
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 $2, 3, \and 5$.
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!
Solve the Question
Alright, first to get $a$. First let’s test $22$. $22$ is even, so it’s divisible by $2$. Next is $23$. We don’t recognize it from our $9×9$ multiplication table and $23$ is not divisible by $2$, $3$, or $5$. $23$ is a prime number, and the lowest prime number greater than $21$.
Now for $b$. First let’s test $15$. We recognize it from our $9×9$ multiplication table as being $15=3·5$. Next, we know that $14$ is divisible by $2$ since it is even. Now for $13$. We don’t recognize it from the $9×9$ multiplication table. Also, it’s not divisible by $2$ since it’s odd. $13$ isn’t divisible by $3$ since the sum of its digits, $4$, is not divisible by $3$. And we know it’s not divisible by $5$ since it doesn’t end in $0 \or 5$. So $13$ is our greatest prime number less than $16$.
Multiplying $23$ and $13$ together on our calculator we get $23·13=299$. So The correct answer is A, $299$.
What Did We Learn
This question was all about how quickly we could determine if a number is prime. Memorizing the $9×9$ multiplication table can speed up this process, as well as memorizing the divisibility rules for $2, 3, \and 5$.
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