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

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

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18 Jul 2019, 08:00
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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

 This question was provided by Veritas Prep for the Game of Timers Competition

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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, 08:07

because 12 unique factors ...in answer choice, 11 is the larger prime number.
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Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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Updated on: 18 Jul 2019, 23:06
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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.

Originally posted by Sayon on 18 Jul 2019, 08:14.
Last edited by Sayon on 18 Jul 2019, 23:06, edited 1 time in total.
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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, 08:15
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$$12=2*2*3$$

So $$N=p1^a*p2^b*p3^c$$ where $$p1, p2 and p3$$ are distinct prime number and one of $$a, b and c$$ can be 2 while the other two are 1

This is because if $$N$$ had 4 unique prime factors, $$N=p1^1*p2^1*p3^1*p4^1$$ the number of factors becomes 16

So N can have a maximum of 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, 08:17
12 unique factors means 11 prime factors + 1 as factor.
therefore 11 unique prime factors.

Hence D
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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, 08:18
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Total unique factors => 12= (2)*(2)*(3)...To maximize the count of prime factors
12= (1+1) * (1+1)* (2+1), so there can be three distinct prime numbers that are also factors of N.
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Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 08:19
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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

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
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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, 08:23
3
N has 12 factors so it isn't a perfect square.

Total number of factors of N= a^m * b^n *... is calculated as-
(m+1)*(n+1)*..
To get 12 as total factors, largest possible number of unique prime factors is 3-
a^1*b^1*c^2

=> (1+1)*(1+1)*(2+1) = 12

Hence 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, 08:25
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?

(A) 2

12=2ˆ2*3; so 2 unique 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, 08:25
1

N = a1^x*a2^y*a3^z....
number of factors = (x+1)*(y+1)*(z+1)...
as number of factors are 12 = 2*2*3
we can represent N as a1*a2*a3^2, where a1,a2 and a3 are primes, so 3 primes

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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, 08:33
1
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

SOL:

if there are 12 unique factors then we know that the number of factors= (power of unique prime +1)(power of other unique prime +1)(.... so on)

so we know that 12 has following factors

2*6 (1 prime has power 1 and other prime has power 5)
2*2*3(2 primes have power 1 and 1 prime has power 2)

only the above combination is possible.

so we can safely say that 3 is the correct answer.

why not 7 ,11 or 12... heres why,if there are 12 unique primes then all of them has to have power 0 and then all the values will be=1.
similarly for 11 because it will still make atleast 1 prime to have power as 0 then it will be 1 and 1 is not a prime. similarly for 7 atleast one of the prime factors has to have power 0 so it will also not make a prime factor.
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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, 08:34
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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

This question is from the number of properties.
We know that the total number of factors is the multiplication of powers +1 of the prime numbers.
12 = 2*2*3
Which means that it will have 3 uniques 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, 08:34
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Let number N=a^p.b^q.....

Number of factors would be (p+1)(q+1)....

let's break 12 into maximum no of factors greater than 1

12 = 2X2X3
so N= a^2.b^2.c^3

Hence, Maximum no of unique prime 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, 08:34
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N = p1^a*p2^b*p3^c , in which p1,p2, and p3 are different primes and a,b, and c are the exponents of those different primes
Number of factors of N= (a+1)(b+1)(c+1)=12
12 could be the multiple of 2*6, 3*4, or 3*2*2, each of these represents the exponent+1 of one prime; so with more exponents we can maximize the number of primes within N.

The max number of prime we can have is 3.

Please note that 1*2*2*3 doesn't work. This would mean p1^0*p2^1... so p1 is not factor of N
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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, 08:43
1
B

This looks like a very straight forward question unless I am missing something which I feel I am. Hopefully I am wrong.

N has 12 unique factors, which means no of factors of N is 12. Note that factors of 12 are 2*2*3. So, N could be, N = 2^3*3^2 or N = 2*3*5^2. The no of prime factors in these two cases are 2 and 3, respectively.

So maximum possible factors are 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, 08:43
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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

Factorization formula is a power of the prime number +1 and then multiply them.
We should have as many prime numbers as we can, so 2*2*3
3 is the largest possible number of prime numbers

IMO 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, 08:47
If N>0 and has 12 unique factors, then the maximum number of prime factors N can have is also 12. Consider: $$N= 2*3*5*7*11*13*17*19*23*29*31*37$$, N here has 12 unique factors all of which are prime.

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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, 08:48
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IMO correct answer is B - 3; explanation is provided as attachment
Attachments

IMG_20190718_211608.JPG [ 949.98 KiB | Viewed 1436 times ]

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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, 08:48
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

N has exactly 12 unique factors

12 can be written as
12*1
6*2
3*4
2*2*3

No. of unique factors = sum of the power of prime factors+1

So, to maximize the number of unique factors we should choose 2*2*3 option.
N=p1*p2*$$p3^2$$

Hence we get 3 prime factors.

If we could have split the number into 4, we would have chosen that option.

Hence, Ans should be (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, 08:52
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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

num.factors of an integer x is the product of the powers of the "prime factors of x + 1"
test some cases num.f(N)=12, we find that the max is 3:
N=2ˆ11=(11+1)=12 <= 1 prime
N=2ˆ3*3ˆ2=(3+1)(3)=12 <= 2 primes
N=2ˆ2*3*5=(2+1)(2)(2)=12 <= 3 primes
N=2*3*5*7=(2)(2)(2)(2)=16 cant be.

Re: Positive integer N has exactly 12 unique factors. What is the largest   [#permalink] 18 Jul 2019, 08:52

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