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Positive integer N has exactly 12 unique factors. What is the largest
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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
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 08:07
Answer D: 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
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Updated on: 18 Jul 2019, 23:06
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 factors – This one being the largest.
Answer B
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
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18 Jul 2019, 08:15
\(12=2*2*3\)
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
Answer is (B)



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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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
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18 Jul 2019, 08:18
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
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18 Jul 2019, 08:19
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
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18 Jul 2019, 08:23
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
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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
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18 Jul 2019, 08:25
IMO answer is B
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
answer B



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 08:33
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
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18 Jul 2019, 08:34
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.
Hence the answer is B.



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 08:34
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
B is the Answer.



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 08:34
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
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18 Jul 2019, 08:43
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
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18 Jul 2019, 08:43
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
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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.
Answer is E.



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 08:48
IMO correct answer is B  3; explanation is provided as attachment
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 08:48
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
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18 Jul 2019, 08:52
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. Answer (B).




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