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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 11:31
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
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18 Jul 2019, 11:31
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.
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, 11:36
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
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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
Answer Choice : 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, 12:22
It could have 12 prime numbers as factors Posted from my mobile device
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 13:26
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
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, 13:50
Answer E: 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
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18 Jul 2019, 14:11
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
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, 14:26
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
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18 Jul 2019, 14:42
Answer is ASince 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 Therefore answer is A
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 15:34
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.
Therefore, the answer is B



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Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 17:34
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
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18 Jul 2019, 18:33
\(N\) 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 nonprime ones.
Answer is (B)



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Positive integer N has exactly 12 unique factors. What is the largest
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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
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18 Jul 2019, 20:12
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. Then answer is therefore 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, 21:16
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 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, 21:20
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
Correct 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, 21:30
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
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18 Jul 2019, 21:53
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
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18 Jul 2019, 21:59
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
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