The integer K is positive, but less than 400. If 21K is a multiple of

Question

The integer K is positive, but less than 400. If 21K is a multiple of 180, how many unique prime factors does K have?

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

Given that 21K/180 is an integer than 7K/60 is also an integer. Therefore, K must be divisible by 60.

Prime factorization of 60 is 2^2 * 3 * 5 resulting in 3 unique prime factors.

KarishmaB · Accepted Answer

Also note here the relevance of 'K must be less than 400'. 
21K is 180n (a multiple of 180).

180n = 2^2 * 3^2 * 5 * n = 3*7*K

n must have a 7 at least.

K must have two 2s, a 3 and a 5 at least. This means it must be at least 2*2*3*5 = 60. 
So K has the following prime factors: 2, 3 and 5. Can it have any other prime factors? The next smallest prime factor is 7. But 60*7 = 420 - a number greater than 400. This means that if K is greater than 60, the only other prime factors that K can have must be out of 2, 3 and 5 only. That is, K may be 60*2 or 60*3*2 or 60*5 etc. This tells us that K has exactly 3 prime factors. If we did not have this condition of K less than 400, we would not know exactly how many factors K has.

Answer (C)

Carcass · Answer

3*7*k/2^2*3^2*5

From this follow that 7*k/2^2*3*5. Now 7 to numerator and 2 - 3 - 5 to denominator share nothing so to be an integer our fraction K must have a 2 a 3 and a 5 , no matter what.

hence our answer is C. three factors

PareshGmat · Answer

\frac{21k}{180} = \frac{7k}{60}

k can be 60, 120, 180, 240, 360

60 = 2^2 * 3 * 5

Unique factors are 2, 3 & 5

Answer = C = 3

elizaanne · Answer

You just have to pick one value for k that works, they must all have the same number of unique primes, otherwise the problem wouldn't work. So, try to pick the simplest one. I picked 60, because 21k, is essentialy 3*7*k and k*60=180, so 21*60 is a multiple of 180.

if we break 60 into its primes, we get 2*2*3*5, the problem is looking for unique primes, so the answer is 3 (C)

if we break 60 into its primes, we get 2*2*3*5, the problem is looking for unique primes, so the answer is 3 (C)

BrainLab · Answer

21k=180n 
k=3*3*2*2*5*n/7*3 so if we want k to be an integer n must be a multiple of 7 but max 42=2*3*7 because otherwise K>400
SO we have 3 unique prime factors 2,3,5 Answer (C)

blueciti · Answer

Hi Elizaanne,

I have a question regarding your approach. I understand the answer is C.

You just have to pick one value for k that works, -  I totally understand. No problem here.

they must all have the same number of unique primes, otherwise the problem wouldn't work . -  How can i know that

In the above question we have 21K is a multiple of 180

that is 21K/180 is an integer

21K/180 can be written as 7*3*k/2^2*3^3*5

From the above equation, I can say k must have 2,3,5. At this point, K can have other prime factors as well
Upon checking other values for k (360,180 ....) ( given in question 400>K>0), i can conclude that K has 3 distinct prime factors i.e. 2,3,5

So, try to pick the simplest one. I picked 60, because 21k, is essentialy 3*7*k and k*60=180, so 21*60 is a multiple of 180.

How did you conclude that 60 will include all the distinct prime factors?

Thanks in advance

GMATMBA5 · Answer

21k is multiple of 180, that is, 180*n=21k
k=(180*n)/21 = (60/7)*n.

It is also given that k is an integer, so n has to be multiple of 7, which also means that k has value in multiples of 60.
Unique prime factors of 60 are 2*3*5.

We need to check if k can have more unique prime factors. It is also given that 0<k<400. So this way the max value of k we have from above is 360, which also has same unique prime factors.
So the answer is C.

ScottTargetTestPrep · Answer

Since 180 = 18 x 10 = 3^2 x 2^2 x 5 and 21K = 3 x 7 x K, we see that K must be a multiple of 3 x 2^2 x 5.

In other words, K = 3 x 2^2 x 5 x n = 60n for for some positive integer n. We see that K already has 3 unique prime factors, namely, 2, 3 and 5. However, since K is less than 400, we see that n can’t be more than 6. Because n is no more than 6, we see that K can’t have any more prime factors other than 2, 3 and 5.

Answer: C

elchaya · Answer

Why did they use in the question the word (( UNIQUE )). why the prime 2 is considered a unique factor, knowing that K shall contain 2^2 ???

BrushMyQuant · Answer

↧↧↧ Weekly Video Solution to the Problem Series ↧↧↧

https://www.youtube.com/watch?v=rZDjHJTJSXw

Given that The integer K is positive, but less than 400 and 21K is a multiple of 180. And we need to find how many unique prime factors does K have?

21K is a multiple of 180
=> 21K is divisible by 180
21K = 3*7*K and 180 = 3*60

So, for 21K to be divisible by 60
K should be divisible by 60

=> K is a multiple of 60 (=2^2 * 3 * 5)
Unique prime factors of K = 3

So,  Answer will be C 
Hope it helps!

Watch the following video to learn the basics of Factors and Multiples

https://www.youtube.com/watch?v=1rw-WLkavcM

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The integer K is positive, but less than 400. If 21K is a multiple of

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The integer K is positive, but less than 400. If 21K is a multiple of

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards