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?

A) 3
B) 6
C) 7
D) 8
E) 10

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.

Setup

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.

Solve

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.

Setup 2

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)

Solve 2

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.

 

 

If n = (33)^43 + (43)^33 what is the units digit of n? GMAT Explanation, Video Solution, and More Practice!

If n = (33)^43 + (43)^33 what is the units digit of n?

A. 0
B. 2
C. 4
D. 6
E. 8

Units digits questions always come up in tutoring. They're great because they're very formulaic (unlike a lot of the GMAT). I remember really struggling for time on an official GMAT I was taking and then question 31 was: What's the units digit of 8^52? Blasted through that one is 20 seconds and: 770! I was psyched! Anyways, rest assured that GMAT units digits questions are easy.

Every units digit has a multiplication pattern with the longest patterns being only 4 variations long. So let's look at some units digit patterns and then we'll apply what we learn to the question at hand. With each of these we're just raising the power to get to the next number in the pattern:

1: Always ends in 1

2: 2, 4, 8, 6 (2^1, 2^2, 2^3, 2^4). Then you can see that 2^5 goes back to "2".

3: 3, 9, 7, 1

4: 4, 6

5: Always ends in 5

6: Always ends in 6

7: 7, 9, 3, 1

8: 8, 4, 2, 6

9: 9, 1

0: Always ends in 0

You don't have to memorize these! They are easy enough to derive in the moment. An important note when creating units digit patterns: you don't have to multiply the entire number to yield the units. You just need to multiply the units digits. So 37*37, no need to multiple the whole thing. Just take 7*7 and you know that number ends in 9.

Ok, easy right? Let's get back to: If n = (33)^43 + (43)^33 what is the units digit of n?

Let's figure out the units for each one and then, guess what, you just add those units digits. Adding and subtracting units digits is also do-able without having to deal with the entire number.

You might be wondering how we're going to get to the 43rd number in the pattern for (33)^43. Well, that's just a question of divisibility. The units is 3 so we have a pattern of 4 numbers. How many 4's fit into 43? 10 with 3 leftover. The leftover is the important part.

3

9

7

1

That 3 means we end in 7. There are 40 complete patterns of 4 and then 3, 9, 7 gets you to 43. Put another way, 7 is the 43rd number in the pattern.

Same thing for 33. There are 32 complete patterns of 4 with 1 leftover. So that leaves a units digit of 3.

You can test this all out by just writing out 3, 9, 7, 1 a bunch and counting to the 33rd and 43rd terms. You will find 1 and 7 respectively. But again, easy way to figure out where you are is to divide the exponent by the number of numbers in the pattern.

Then just add the units to yield the units of the sum of the numbers: 7 + 3 = 10. So the units digit of n is 0.

Correct Answer: A

If n equals 33 43 + 43 33 what is the units digit of n? GMAT Explanation Diagram

Video Solution: If n = (33)^43 + (43)^33 what is the units digit of n?

Additional GMAT Units Digit Practice Questions

Here is a very similar units digit question from GMAT Question of the Day but it has an important twist. Make sure you follow through with all of you basic math rules.

This GMAT question of the day puzzle goes beyond digits into powers of 10 but is a great one to be familiar with as I've seen this style on a multiple official GMATs.

Here's a digits Data Sufficiency Question from GMAT Question of the Day It's a bit more involved but good practice.

The product of all the prime numbers less than 20 is closest to which of the following powers of 10? GMAT Explanation, Video Solution, and More Practice!

The product of all the prime numbers less than 20 is closest to which of the following powers of 10?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

I really like this question. The type of thinking required for success here is exactly what you need to succeed overall on the GMAT and what we try to reinforce each and every tutoring session.

Prime numbers less than 20

Let's list out those numbers. Take your time. Double check. This isn't a race. Or at least it isn't a sprint. You'll end up saving time by doing work in a careful and considered way.

2, 3, 5, 7, 11, 13, 17, 19

Just a quick note on primes. 2 is the only even prime and the smallest prime. Important thing to have memorize for the GMAT.

Ok, so we're looking to match a power of 10 with the product of these numbers. A power of 10 is 10*10*10... some number of times. So let's figure out a way to translate the product of the those primes to powers of 10. Start easy. You don't have to figure out everything all at once.

2*5 = 10. Great, we've got one 10.

3*7 = 21 which is basically 2*10

11 is basically 10

13 is a bit of a stretch but let's call it 10 for now and judge whether it's OK once we've got the first round done.

17 let's call 20 and maybe it balances out 13.

19 let's call 20 and it balances out 11.

So we've got 10*10*10*10*10*10*2*2*2*2. So 10^6*8. That's closest to 10^7.

Now, I get that you might be skeptical of the approximation BUT we did a reasonable approximation and 10^6*8 is way closer to 10^7 than 10^6. It's not close. In GMAT land I think that's good enough and I'd pick C and move on.

There is an underlying principle that proves this without a doubt that's worth knowing.

Consider:

2*100 = 200

2*101 = 202

3*100 = 300

When we added 1 to 100 we moved from 200 to 202.

When we added 1 to 2 we moved from 200 to 300.

So adding a fixed value to smaller number creates a bigger change. That makes sense since the fixed value increases the smaller by a greater percent than it increases the bigger number.

Back to our question!

2*5 = 10

3*7

11 (-1) vs 19 (+1) Given the above idea which move has more influence? The smaller one, 11.

13 (-3) vs 17 (+3) Same thing here. 13 has more influence.

So with those two we 100% underestimated. And then with 7*3 we also went down to 20. So it's clear that our approximation was even smaller than the actual number guaranteeing that the product of all the primes less than 20 is closest to 10^7.

Correct Answer: C

The product of all the prime numbers less than 20 is closest to which of the following powers of 10? GMAT Explanation Diagram

Just a quick note beyond this question. So we showed above that when adding a fixed value to a number the magnitude of the number you're adding to changes the affect of the number you add (adding a fixed value to 2 will have more impact on 2 than on a larger number say 100000).

Multiplication is different. Multiplying something by 2 doubles it's value regardless of the original magnitude (2*2 is 4, 2*100 is 200). Of course you know that. But it's important on the GMAT to put this all together in a meaningful way. We call it "absolute vs relative" and it comes up a decent amount especially on the GMAT Data Sufficiency. It tends to be that relative info (multiplication) which stays constant regardless of the magnitude of the underlying numbers is usually more helpful than absolute info, addition/subtraction.

Video Solution: The product of all the prime numbers less than 20 is closest to which of the following powers of 10?

Additional GMAT Puzzle Question Practice

Not the same but requires similar GMAT-puzzle-thinking. Here's a GMAT Question of the Day puzzle dealing with exponents and primes

Another similar but different puzzle question. This GMAT Question of the Day Exponents Puzzle deals more with constraints but again still tests the same type of organization skills that lead to success in the above primes question.

 

How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5? GMAT Explanation

How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5?

A. 1
B. 2
C. 3
D. 4
E. 5

If you start doing the algebra, working the inequality/quadratic, this can get ugly. On GMAT quant remember to stay practical. Not everything has a tidy algebraic solution. Some things do so let's not completely forget about solving equations/inequalities but, again, let's just make good decisions and use the tools that make sense for the job. In this case we have a single variable inequality and a somewhat limiting constraint for the value of x: How many of the integers...less than 5.

My gut would be to start testing numbers less than 5: 4, 3, 2, 1, 0, -1, -2, -3...

The inequality we're attempting to satisfy, (x+2)(x+3)/(x−2) ≥ 0, hinges on the expression being positive or negative. With that in mind I'd pay special attention to signs. We don't really care about the actual value of the expression just whether we are above or below zero. Context is key!

Here's a video if you need to brush up on multiplying positive and negative numbers. If though you are missing that fundamental you might want to take a step back in your GMAT prep and dig back into the basics. This is a pretty advanced question to tackle if you're having trouble with signs.

OK - back to work! Popping in anything 3 or greater is positive so 4 and 3 are good. 2 yields a zero in the denominator so that's not going to work because diving by zero is undefined. -1 yields negative. -2 and -3  each yield zero so both work. -4 yields negative. So does -5 and everything smaller than that. So -2, -3, 3, and 4 all satisfy the inequality. D. 

How many of the integers that satisfy the inequality (x+2)(x+3):(x−2) ≥ 0 are less than 5? GMAT Explanation Diagram

Video Explanation: How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5?

Additional Algebra, Inequality, Positive/Negative, Picking Numbers GMAT Practice Questions

This GMAT Question of the DS Number Properties/Signs example is a bit different in the specifics especially because it's Data Sufficiency but the focus on positive/negative is spot on.

Here's the another Data Sufficiency example question with testing signs. Again, a little different but very similar in certain important aspects.

More practice dealing with signs this time with absolute value thrown in the mix.

A school administrator will assign each student in a group of N students to one of M classrooms. If 3 < M < 13 < N, is it possible to assign each of the N students to one of the M classrooms so that each classroom has the same number of students assigned to it?GMAT Explanation

A school administrator will assign each student in a group of N students to one of M classrooms. If 3 < M < 13 < N, is it possible to assign each of the N students to one of the M classrooms so that each classroom has the same number of students assigned to it?

(1) It is possible to assign each of 3N students to one of M classrooms so that each classroom has the same number of students assigned to it.

(2) It is possible to assign each of 13N students to one of M classrooms so that each classroom has the same number of students assigned to it.

This is a super tricky GMAT number properties/divisibility question from the GMAT official guide. Oddly we don't see this one a lot in GMAT tutoring sessions but maybe we can credit the video explanation for that. As always let's start by reading carefully and figuring out the question: is it possible to assign each of the N students to one of the M classrooms so that each classroom has the same number of students assigned to it? OK, we know that if it has a question mark at the end it's a question BUT what does that actually mean? Especially on Data Sufficiency you really need to do a step further. In this case the question is, does M divide evenly into N. Or, is N divisible by M. Or is N/M an integer. However way you want to look at it. The point is: dig into Data Sufficiency information and try to make an inference, organize, or transform in some way. Dig!!!

OK, so now we've got the question, and if we had the values for M and N it would be very simple. Of course, we don't have the values but in the statements we'll be given constraints which will potentially narrow things down to sufficiency. We do have a constraint in the questions stem though: M is between 3 and 13 and N is greater than 13. For M that's quite a narrow range so I might just write values out: 4, 5, 6, 7, 8, 9, 10, 11, 12. Boom! Easy organization.

Let's get to the statements:

(1) It is possible to assign each of 3N students to one of M classrooms so that each classroom has the same number of students assigned to it.

What does that mean??? 3N/M is an integer. How does that relate to N/M. Let's try to do a Yes/NO.

M = 4, 5, 6, 7, 8, 9, 10, 11, 12  N > 13

Yes

Pick a value for M and then pick something for N in the numerator that will cancel out that M (pick an N that's divisible by the M).

M = 4

N = 40

Make sure you're keeping N and M constraints in mind.

3*40/4 = integer

40/4 = integer

 

No

For the no we'll do something similar but this time pick an N not divisible by the M. Let's pick an M that will be at least partially cancelled out by the 3. So 6 would be a good M. And then just pick an even N (to cancel the 2 in 6) not divisible by 6. So N could be 14.

3*14/6 = integer

14/6 is not equal to an integer.

For statement one we have a yes and a no so it's insufficient.

(2) It is possible to assign each of 13N students to one of M classrooms so that each classroom has the same number of students assigned to it.

M = 4, 5, 6, 7, 8, 9, 10, 11, 12  N > 13

Statement two looks very similar BUT just because statements look similar doesn't mean they''ll have the same outcome. The details matter. Still, we can use the same procedure setting up a yes/no to test sufficiency.

Yes

So 13N/M = integer

So M could be 7 and N 14.

13*14/7 = integer

No

Given the constraint this is impossible. 13/M will never be an integer because M can only be 4, 5, 6, 7, 8, 9, 10, 11, 12. Eh? So what! Well, if M can't be cancelled out by the 13 then how is it going to disappear from the denominator to leave an integer? The N has fully cancel the M. Aha! So, given this constraint N/M must be an integer. Statement two is sufficient.

B.

A school administrator will assign each student in a group of N students to one of M classrooms GMAT Explanation

Additional GMAT Data Sufficiency Divisibility Practice Questions

GMAT Data Sufficiency Divisibility Question of the Day 1

GMAT Data Sufficiency Divisibility Question of the Day 2

GMAT Data Sufficiency Divisibility Question of the Day 3

 

GMAT ARTICLES

CONTACT


Atlantic GMAT Tutoring

405 East 51st St.

NY, NY 10022

(347) 669-3545