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

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 11:31
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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

total no of factors is 12.

so this can be arrived by possibilities
1X 12
4 X 3
6 X 2
2 x 2 x 3

out of which the maximum possibility is 2 * 2 * 3 which says 3 prime numbers are needed. so ans is 3 ie B
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Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 11:31
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Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?

We can write 12 as following:

12 = 12*1 = 6*2 = 4*3 = 2*2*3

Based on the formula for total unique factors, we can write N in the form of powers of prime factors of N.

N = a^11 = (b^5) * c^ = (d^3) * (e^2) = f*g*(h^2) ......................(Here a b c, ...h are some prime numbers)

N can have any of these forms and still have 12 unique factors. But, the form in the boldface will give N the largest possible number of unique prime factors, which will be f, g & h. Total 3 unique prime factors.

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 11:36
1
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

12 no of factors having max prime no = a^2 ,b^1,c^1 or a^1 ,b^2 ,c^1 or a^1,b^1 & c^2.....(where a,b&c are prime nos)
(such that, factors= 3*2*2=12)

Thus max no of unique prime factors =3(a,b &c )

Ans B
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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 11:58
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?

Positive integer N has exactly 12 unique factors.
If N = 1*2*3*4*...*11*12
then largest possible number of unique prime factors that N could have = 5 (prime factors being 2,3,5,7,11)

Similarly, If all 12 factors are prime numbers then largest possible number of unique prime factors that N could have = 12

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

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 12:22
It could have 12 prime numbers as factors

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 13:26
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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

First of all, let's recall a theory. Rather boring topic, necessary one.
We know that any number may be presented as the product of prime factors raised to some power greater of equal to 1. Sounds like a song!
To find the number of divisors we need to take powers of these prime factors, add 1 to each and multiply: ?=(?1+1)(?2+1)…(??+1), where m is the number of prime divisors.

In our case we have 12 unique divisors.
12 = 2*2*3 = (1+1)*(1+1)*(2+1)

So the max number of prime factors is 3

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 13:50
If N has unique factors, it could have all prime factors only as its factor, So highest number is possible
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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 14:11
1
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?

List of numbers with exactly 12 unique factors
.: N could be :
60 = 2^2•3•5 (three prime factors)
72= 2^3•3^2 (two prime factors)
84= 2^2•3•7 (two prime factors)
90 = 2•3^2•5 (three prime factors)
96= 2^5•3 (two prime factors )
486= 2•3^5 (two prime factors)
2048= 2^11 (one prime factor)

So clearly ,the largest possible number of prime factor that N could have is 3

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 14:26
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according to different factorizations of 12, N can have several forms (where a,b,c are different primes):
12*1 --> $$N = a^{11}$$ --> 1 prime
6*2 --> $$N = a^5b^1$$ --> 2 primes
4*3 --> $$N = a^3b^2$$ --> 2 primes
3*2*2 --> $$N = a^2b^1c^1$$ --> 3 primes , so B
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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 14:42

Since positive integer N has exactly 12 unique factors, lets try with a number that has 12 unique factors
2^3*3^2
=8*9
=72; 12 Unique factors of 72 are: 1,2,3,4,6,8,9,12,18,24,36, and 72
Here we can see that 72 has only 2 unique prime factors and that is 2 and 3.

Lets try this with another number that has unique 12 factors
3^3*2^2
=27*4
=108; 12 Unique factors of 108 are:1,2,3,4,6,9,12,18,27,36,54, and 108
Again we can see that 108 has 2 unique prime factors and that is 2 and 3

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 15:34
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Let's say N = a^x * b ^y * c^z.... where a,b,c is prime factor
The total unique factor is calculated by (x+1) * (y+1) * (z+1)..., which is equal to 12.

minimum value of x, y, z, etc. is 1.

For 3 prime factors, the lowest total unique factor is 2^3 = 8 < 12.
For 4 prime factors, the lowest total unique factor is 2^4 = 16 > 12.

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

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18 Jul 2019, 17:34
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12 can be written as 1*12, 2*6, 3*4 or 2*2*3.

Number of unique factors of N= a^x*b^y*c^z.... are (x+1)*(y+1)(z+1).., where a, b and c are prime numbers.

Hence, Positive integer N has exactly 12 unique factors can be written as
i) $$a^{11}$$
ii) $$a^1*b^5$$
iii) $$a^2*b^3$$
iv) $$a^1*b^1*c^2$$
where a, b and c are prime numbers.

Hence largest possible number of unique prime factors of N is 3

IMO B
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Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 18:33
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$$N$$ has exactly 12 unique factors. Several possible scenarios of $$N$$ are as follows:
(1)$$N = a*b^5$$, where $$a$$ and $$b$$ are both positive prime numbers
(2)$$N = c^2*d^3$$, where $$c$$ and $$d$$ are both positive prime numbers
(3)$$N = k*l*m^2$$, where $$k, l,$$ and $$m$$ are all positive prime numbers

Basing on above scenarios, the largest possible number of unique prime factors that N could have is $$3$$ .
e.g. $$60=5*3*2*2$$, has 12 unique factors: $$1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 -$$3 unique prime factors and 9 non-prime ones.

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

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Updated on: 19 Jul 2019, 20:57
total factors of N = 12
so largest possible unique prime factors of N ; 3
IMO B

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
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Originally posted by Archit3110 on 18 Jul 2019, 19:04.
Last edited by Archit3110 on 19 Jul 2019, 20:57, edited 1 time in total.
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Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 20:12
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If N has prime factorization of (a^x)*(b^y)*(c^z)*...*(d^t), then the unique factors of N =(x+1)*(y+1)*(z+1)*...*(t+1)
where a,b,c,d are prime numbers and x,y,z,t are positive integers greater than zero. The minimum value of x,y,z,t is 1.
Hence, N which has 12 can not have 4 prime factors
Since
12<(1+1)(1+1)(1+1)(1+1)
12<16.
However 12>(1+1)(1+1)(1+1)
i.e. 12>8 hence it means N has 3 prime factors with powers x,y,z such that x=y=1 and z=2
Implying 12=(1+1)(1+1)(2+1)
12=12.

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 21:16
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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

12 unique factors
using formula
12 = 2*2*3 (breaking into prime factors)
thus 12 will be of the form $$a^1*b^1*c^2$$
since the 12 = (1+1)(1+1)(2+1)
hence maximum number of prime factors will be 3
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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 21:20
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Total # of factors of any number is given by (x+1)(y+1)(z+1)..., where x, y, z are the powers of the unique prime factors

If # of factors = 12, --> 4*3 or 2*6, which in turn could be 2*2*3. That means, 3 unique prime factors each of them with powers 1, 1 and 2.

Hence, max possible number of unique factors = 3

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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 21:30
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Quote:
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

12 can be split into 2*6 or 3*4
but if we want largest possible number of unique prime factors the then we need to split it to the lowest possible form i.e 2*2*3

therefore N can be $$a^1 *b^1* c^2$$
where a,b,c are prime factors

hence 3, option B
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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 21:53
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Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?

N can be in power of 11 ---> one prime factor
N can be $$x^5$$ and $$y^1$$ = (5+1)*(1+1)=12 (x and y are primes) ----> 2 prime factors
N can be $$x^1$$ *$$y^1$$ * $$z^2$$=(1+1)*(1+1)*(2+1)=2*2*3=12 (x, y, and z are prime) -----> 3 prime factors
Maximum number of unique prime factors is 3 (B)
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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 21:59
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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

Sol:

For a number to have 12 factors, it must be of the form a^1 * b^1 * c^2, where a, b, and c are prime numbers.
Hence, it can have a maximum of 3 prime factors.
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Re: Positive integer N has exactly 12 unique factors. What is the largest   [#permalink] 18 Jul 2019, 21:59

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