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# Positive integer N has exactly 12 unique factors. What is the largest

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Manager
Joined: 12 Mar 2019
Posts: 111
Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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19 Jul 2019, 23:30
Kinshook wrote:
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 12

No of unique factors of positive integer N = 12
12 = 2*2*3 = 4*3 = 2*6 = 12
Therefore

$$N = P_1*P_2*P_3^2$$ for 12 = 2*2*3. No of unique prime factors = 3 (1)
or
$$N = P_1^3*P_2^2$$ for 12 = 4*3 No of unique prime factors = 2 (2)
or
$$N= P_1*P_2^5$$ for 12 = 2*6 No of unique prime factors = 2 (3)
or
$$N= P_1^11$$ for 12 = 12 No of unique prime factors = 1 (4)

We see that for equation (1), no of unique prime factors = 3 is largest possible number of unique prime factors

IMO B

Hello ,
I have small doubt as i am new to this concept. I understand rule of finding factors : power +1 : But question stem says N has 12 unique factors. In all cases you presented, if P2^2 or P^3 it means, its p2*p2 or p3*p3*p3, How can we take this as question says it has Unique factors, but from this we have 2 times P2 and 3 times P3, which cannot be unique as its repeating. Don't condition itself falls at time of assumption as it doesnot have unique 12 factors, Can't it be p1,p2,p3----p12 .
Hope i am able to explain my question
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Schools: HBS '14 (A)
GMAT 1: 730 Q51 V36
Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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19 Jul 2019, 23:42
rishab0507 wrote:
Kinshook wrote:
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?

(A) 2
(B) 3
(C) 7
(D) 11
(E) 12

No of unique factors of positive integer N = 12
12 = 2*2*3 = 4*3 = 2*6 = 12
Therefore

$$N = P_1*P_2*P_3^2$$ for 12 = 2*2*3. No of unique prime factors = 3 (1)
or
$$N = P_1^3*P_2^2$$ for 12 = 4*3 No of unique prime factors = 2 (2)
or
$$N= P_1*P_2^5$$ for 12 = 2*6 No of unique prime factors = 2 (3)
or
$$N= P_1^11$$ for 12 = 12 No of unique prime factors = 1 (4)

We see that for equation (1), no of unique prime factors = 3 is largest possible number of unique prime factors

IMO B

Hello ,
I have small doubt as i am new to this concept. I understand rule of finding factors : power +1 : But question stem says N has 12 unique factors. In all cases you presented, if P2^2 or P^3 it means, its p2*p2 or p3*p3*p3, How can we take this as question says it has Unique factors, but from this we have 2 times P2 and 3 times P3, which cannot be unique as its repeating. Don't condition itself falls at time of assumption as it doesnot have unique 12 factors, Can't it be p1,p2,p3----p12 .
Hope i am able to explain my question

When we say N = a^p * b^q, then we mean that N has (p + 1) * (q + 1) unique factors.

For example 72 = 2^3 * 3^2
Hence, 72 has (3 + 1) * (2 + 1) = 12 unique factors.
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
As you can see, there are 12 unique factors and nothing is repeated.

Hope this helps.
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Senior Manager
Joined: 13 Feb 2018
Posts: 448
GMAT 1: 640 Q48 V28
Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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19 Jul 2019, 23:56
rishab0507

the stem says "unique factors" not "uniques prime factors"

You Assert: " its p2*p2 or p3*p3*p3, How can we take this as question says it has Unique factors, but from this we have 2 times P2 and 3 times P3,"
1) you have to brush basic concepts: $$p^2=2*p$$ is wrong
2) consider p=2, $$p^3=2*2*2=2^3=8$$ You assert that 2*2*2 "cannot be unique as its repeating";
2^3=8 has 4 unique factors, even 2 is repeating: 1, 2, 4, 8

In your example: "p1,p2,p3----p12" this expression has 12 "unique prime factors" and $$2^12$$ "unique factors"

Hope I understood your question well
Intern
Joined: 14 Oct 2018
Posts: 16
Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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22 Jul 2019, 00:28
Sayon wrote:
Positive integer N can have exactly 12 unique factors in any of the following ways:

12*1 -> (11 + 1) -> $$A^11$$ -> 1 prime factor
6*2 -> (5+1)(1+1) -> $$A^5*B^1$$ -> 2 prime factors
4*3 -> (3+1)(2+1) -> $$A^3*B^2$$ -> 2 prime factors
2*2*3 -> (1+1)(1+1)(2+1) ->$$A^1*B^1*C^2$$ -> 3 prime factorsThis one being the largest.

I get it how you are coming up with (1+1)(1+1)(2+1) = 12 and hence, 3 prime numbers i.e, 1,1 & 2. But 1 is neither prime nor composite. So, should it not be (3+1)(2+1), 2 prime numbers, 3&2?

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Re: Positive integer N has exactly 12 unique factors. What is the largest   [#permalink] 22 Jul 2019, 00:28

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