If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following? GMAT Explanation
If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following?
Correct Answer: E
Define the Question
x must be a multiple of which of the following? What does that actually mean? We need to figure out the basic divisibility properties of x and then use that to eliminate answer choices. Do you we need to solve for x? If we could that would be great but we don't need to and probably won't be able to.
There are two nice ways to approach this question. There's a more practical setup and a more number properties approach. Let's go practical first.
We know that Y is a multiple of 5. So why not pick some values for Y? Then we can plug those into the equation and get some possibilities for x and see what inferences we can make.
Let's start at the minimum y multiple of 5 value (considering that both x and y are positive integers), 5, and go from there, 10, 15, 20...
We probably don't more than that.
Now let's plug those y values into the equation: 3x + 4y = 200
3x + 20 = 200
3x = 180
x = 60
3x + 40 = 200
3x = 160
x = 160/3
3x + 60 = 200
3x = 140
x = 140/3
3x + 80 = 200
3x = 120
x = 40
Out of our 4 values of y only 5 and 20 are valid. The others produce non-integer values for x. Let's look at the results for 5 and 20 and see what inferences we can about the divisibility properties of x.
We've 40 and 60. So what's common or the least common multiple? They both have a 5 and a 4. The 3 in 60 and the 8 in 40 are not common so are not a factor that x is must have. Let's look at the answer choices and see what we can eliminate.
A) 3 Not a must have because it's not a factor of 40
B) 6 Not a must have because it's not a factor of 40
C) 7 Not a must have because it's not a factor of 40 or 60
D) 8 Not a must have because it's not a factor of 60
E) 10 With this method we can't prove that this is a must have but we've eliminated everything else and it is common between 40 and 60.
The other way to do this is to simplify the equation and then make some inferences.
3x + 4y = 200
3x = 200 - 4y
3x = 4(50 - y)
Keep in mind that y is a multiple of 5. So again could be 5, 10, 15, 20... What happens when you put any of those numbers in there? 50 - 5 = 45. 50 - 10 = 40. 50 - 15 = 35. You always end up with a multiple of 5. That's because of this divisibility rule:
Div by a number + Div by a number = Remain Div by that number
Div by a number + Not Div by that number = Not Div by that number
Not Div by a number + Not Div by a number = Depends on the number
It's a good rule to know.
So 3x is equal to a multiple of 5 times 4. Meaning: 3x is a multiple of 5 times 4. What about x? Well, the 3 isn't a multiple of 5 or 4 so x must be a multiple of 5 and 4. E is the only option that matches. You might say: but E (10) only has a 2 not a 4! That's OK. We're not being asked: which of the following represents all of the factors of x. We're just asked which answer choice represents factors that x must have. And x must have a 4 and a 5. So it must have a 2 and a 5.
Video Solution: If x and y are positive integers such that y is a multiple of 5 and 3x + 4y = 200, then x must be a multiple of which of the following?
Additional Number Property Divisibility GMAT Practice Questions
Here's another GMAT Number Properties Puzzle question that can be cracked with basic organization: The number 75 can be written as the sum of the squares of 3 different positive integers
And here's a number properties puzzle from GMAT Question of the Day that also benefits from just getting the info from the question in shape.