If x and y are positive integers, what is the value of xy?

If x and y are positive integers, what is the value of xy? GMAT Explanation + Additional Examples!

If x and y are positive integers, what is the value of xy?

(1) The greatest common factor of x and y is 10

(2) The least common multiple of x and y is 180

 

Define the question: What is the value of xy?

There’s not much to define here, but it’s important to note that x and y both are positive integers.

 

Organize the given information with prime factorization: 

Since there’s not too much information from the question, let’s move to the given statements.

 

(1) The greatest common factor of x and y is 10.

Here’s a good refresher on greatest common factors from Khan Academy. Let’s think about this statement in terms of prime factors. The prime factorization of 10 is 2 * 5. From this statement, we know two things:

  1. x and y are multiples of 10 (10 is a factor of both)
  2. The only prime factors x and y have in common are a 2 and a 5.

 

With only this information, x and y can be any number of values — x could be 10 and y could be 20, 30, 40, etc. So, we don’t have a unique answer for xy with only statement 1.

Insufficient.

Let’s look at statement 2. 

 

(2) The least common multiple of x and y is 180.

Here’s another refresher on least common multiples from Khan Academy.

Let’s make a guess that y = 180. Here, x could be 1, 2, 3, or any other factor of 180, as these all have a least common multiple of 180. So again, we can’t find a unique answer for xy using only statement 2. 

Insufficient.

 

Let’s try both statements together.

For this, let’s go back to our prime factorization technique. The prime factorization for 180 is 2 * 2 * 3 * 3 * 5. Both x and y have the factors 2 and 5. So, x and y have to be 10 times some combination of the remaining factors: 2, 3, and 3. 

Let’s write out all the possibilities for x and y: 

10: 2 * 5

20: 2 * 2 * 5

30: 2 * 3 * 5

60: 2 * 2 * 3 * 5 

90: 2 * 3 * 3 * 5

180: 2 * 2 * 3 * 3 * 5

 

Let’s find which combinations of these have a least common multiple of 180. We know two things from statement 2:

  1. Only one of x and y has two 2’s in its prime factorization.
  2. Only one of x and y has 3’s in its prime factorization.

 

This is because x and y only have 2 and 5 as common prime factors. If they both had two 2’s or any 3’s, they would share more prime factors than just 2 and 5. This would violate statement 1. So, what are the possible combinations now?

  1. 10 and 60
  2. 10 and 180
  3. 20 and 90

 

We can rule out option 1 as the least common multiple of 10 and 60 is 60, not 180. However, both options 2 and 3 have a greatest common factor of 10 and a least common multiple of 180.

But, we are looking for the product of x and y. And sure enough these pairs both have a product of 1800. Therefore with both statements together, we can solve for xy, so both statements together are sufficient.

Answer: C

 

Here’s all of that worked out:

If x and y are positive integers, what is the value of xy - Solution