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

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damham17
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The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   23 Dec 2014, 13:43
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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

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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.
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C
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   23 Dec 2014, 22:30
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damham17 wrote:
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

Show Spoiler
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.



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)
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   23 Dec 2014, 13:57
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
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   31 Dec 2014, 00:16
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\(\frac{21k}{180} = \frac{7k}{60}\)

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

\(60 = 2^2 * 3 * 5\)

Unique factors are 2, 3 & 5

Answer = C = 3
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   10 Feb 2015, 00:01
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)
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   08 Oct 2015, 11:58
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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)
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The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   17 Jan 2017, 06:50
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
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   16 Feb 2018, 22:45
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.
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   17 Oct 2019, 20:35
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damham17 wrote:
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

Show Spoiler
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.


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
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   10 Feb 2021, 07:15
VeritasKarishma wrote:
damham17 wrote:
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

Show Spoiler
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.



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)


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 ???
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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink] New post   28 Apr 2023, 12:27
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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

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Re: The integer K is positive, but less than 400. If 21K is a multiple of [#permalink]
28 Apr 2023, 12:27
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