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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, 19:45
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
Here we have that N has exactly 12 unique factors, so N = (P^a)*(Q^b)*(R^c)
The only possibilities to get 12 unique factors are:
1  P^11, total factors =11+1
2  (P^3)*(Q^2), total factors = (3+1)*(2+1) = 12
3  (P^1)*(Q^1)*(R^2), total factors = (1+1)*(1+1)*(2+1) = 12
the third case is the maximum largest possible number of unique prime factors, so (B) is our answer
Originally posted by Mizar18 on 18 Jul 2019, 08:53.
Last edited by Mizar18 on 18 Jul 2019, 19:45, 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, 08:56
Quote: Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have? We are given that \(N>0\) and is integer. It has \(12\) unique factors. We need to find the maximum number of prime factors possible for \(N\) By using factorization formula, \((a+1)*(b+1)*(c+1)=\)Number of factors for a number \(N\), where \(a\), \(b\) and \(c\) are the degrees of different possible prime factors \(A\), \(B\) and \(C\) of number \(N\) We know that number of factors for a number \(N\) is \(12\). Thus, \((a+1)*(b+1)*(c+1)=12\) If we assume that \(a=1\), then \((1+1)*(b+1)*(c+1)=12\) => \(2*(b+1)*(c+1)=12\) => \((b+1)*(c+1)=6\) If we assume that \(b=1\), then \((1+1)*(c+1)=6\) => \(2*(c+1)=6\) => \(c+1=3\) We could potentially add several more prime factors, but in this case we will not get \(3\) in the equation. It means that prime factor \(C\) has a degree equal to \(c=31=2\). Hence, the largest number of prime factors for number \(N\) is \(3\): \(A^1*B^1*C^2\). 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, 08:57
IMO B
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
Given that N has exactly 12 unique factors. And 12 = 2*2*3
We know, if N = a^p*b^q*c^r (where a, b and c are prime numbers and p, q and r are their power respectively) Then the no. of factors N has is equal to (p+1)(q+1)(r+1)
The max no. of factors 12 can have is 3 (as 12=2*2*3)
So N can be written as, N = a^1*b^1*c^2 (refer to above method)
Hence, the largest possible number of unique prime factors that N could have is 3
Please note, 12 can also be written as 2*6, in that case N = a^1*b^5, but we want to maximise the no. of prime factors N could have, hence we have to consider smaller powers for the prime 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, 08:57
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have? If N has 12 unique factors then it can be in the form of \(a^{11}\) or \(a^2\)*\(b^3\) or a*b*\(c^2\) This means the maximum possible number of unique prime factors N can have: 3 (as a*b*\(c^2\) has 3 prime factors which will give total of 12 factors.)
IMO 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, 08:57
No. of factors of any number (x^a*y^b*z^c, where X,Y,Z are prime)will be in the form (a+1)(b+1)(c+1) minimum values a,b and c can take will be 1. So to have maximum number of primes (x,y,z) the values of a,b and c must be minimum. (a+1)(b+1)(c+1)(d+1)...=12 hence 2*2*2*2=16 therefore 4 primes are not possible. 2*2*3=12 hence maximum no. of primes possible is 3IMO 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, 09:03
IMO B.
N has exactly 12 unique factors. Therefore, the largest possible number of unique prime factors that N can have would be to factorise 12.
12 = 1X12 , 2X6 or 3X4. Further 2X6 or 3X4 can be further factorised as 2*2*3. Hence, the smallest number to have 12 factors would be 60 = \(2^2 * 3 * 5\) Therefore, the number of prime factors = 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, 09:05
Let N be any number with exactly 12 factors Picking 2 to obtain N 2^11 = 2048 therefore, 2048 = 1,2,4,8,16,32,64,128,256,512,1024,2048 this means that N is 2048 with 12 unique factors largest possible unique prime factor of N = 2 Hence answer choice AThank you for Kudos.
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 09:05
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
Solution: number of uniq factors in N = 12 = 1*11 = 2*6 = 3*4=2*2*3 So N = \(p^1^0\) or N = \(q^1*r^5\) or N = \(s^2*t^3\) or N = \(u^1*v^1*w^2\), where p, q, r, s, t, u, v &w are prime numbers So maximum number of prime factor is 3 (N = \(u^1*v^1*w^2\))
(A) 2 (B) 3 > correct (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, 09:06
I will go with D
No.can be
2*7*3*5*11*13*17*19*23*29*31
We have 11 prime factors
Don't forget 1 is not a prime factor
D it is !
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 09:07
12 unique factors mean using the number of factors formula, the 12 unique factors can be written as = 6*2 OR 4*3 OR 2*2*3 So the largest possible number of factors = 3 (2*2*3)
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, 09:09
N can be a product of first prime numbers. Imp::: 1 must be one of the factor. Since N has exactly 12 factors. therefore, it can have 11 unique



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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, 01:13
IMOB
N has 12 factors
Note For any number expressed in prime factorization form p^a * q ^b * r^c , Total No of factors of N= (a+1) (b+1) (c+1) [where p, q , r are prime numbers]
12= 1x12 = 2x6 = 3x4 = 3x2x2 A/C N= p^11, or, N= p^5 * q, or, N= p^2 * q^3, or, N= p^2 * q^1 * r^1 [ Where p, q, r are prime Numbers]
For N=p^2 * q^1 * r^1 ,N will have max 3 prime factors p, q, r
Ans 3 B
Originally posted by MayankSingh on 18 Jul 2019, 09:19.
Last edited by MayankSingh on 19 Jul 2019, 01:13, 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, 09:25
We count the number of unique factors as follows: \(N=p^a*q^b*m^c\) ... Where p, q, and m are different prime factors and a, b, c are nonnegative powers of those factors, then the number of unique factors=(a+1)*(b+1)*c+1)
back to our question We know that N has 12 uniques factors. Several cases are possible to get 12: 12*1=12 4*3=12 and so on if we want to increase the number of unique primes in N, we have to split 12 as narrow as possible 2*2*3=12 is the only case (as 2 and 3 are primes and cant be split even more)
So N can have maximum 3 unique prime factors: \(N=p^2*q^2*m^3\) Where p, q, and m are different prime factors
IMO 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, 09:27
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
Any positive integer N can be re written as \(a^x*b^y*c^z\)... and so on where a,b,c... are UNIQUE prime factors and x,y,z... are powers of prime factors.
Number of unique factors of N can be obtained by formula (x+1)(y+1)(z+1)... and so on
N has 12 unique factors
12 = 3x2x2.... 12 can be written as a product of MAX 3 numbers
so N must be \(x^{(31)}*y^{(21)}*z^{(21)}\)... where x, y, and z are unique prime factors
ANSWER: B3 is the largest possible number of unique prime factors 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, 09:29
For 12 unique factors, it should be P1*P2*(P3^2) as this will correspond to total 12 Unique factors ( 2*2*3) Here P1, P2 and P3 are prime numbers.
So there has to be maximum of 3 prime numbers and minimum of one prime number. One can also calculate the total 12 unique factors if there are three maximum prime numbers. 1, P1, P2, P3, P3^2, 1*P1*P2*P3^2 => 6 factors For other combos: there are 4 factors ( P1,P2,P3,P3^2) so unique combinations will be : 3 + 2 + 1 + 0 => 6
So, with 3 prime numbers there are 12 combinations.
Answer > B (3)
Regards, Rishav



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Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 09:59
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have? N = 12 factors = \((no of prime factors)^{11}\) + 1 > which means we should have total number of prime factors to 11 .
We have to maximize number of unique prime factors hence lets say there are 3 prime factors  P, Q, R \(P^1 * Q^1 * R^1\) total prime factors = 2 * 2 * 2= 8 > In order to make this 11, we have to increase the power of either P, Q or R > which means the largest no of prime nos we can have is 3
if I add one more prime no in the mix lets say \(S^1\) which will result in 8 * 2 = 16 total prime factors whereas we are looking for total 12. hence there can be max 3 prime nos  P, Q, R
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, 10:33
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
SOLUTION: As its given that a number hastotal 12 unique factors, so we know that unique factors of any number is given by the formula (e1 +1)*(e2 + 2)*(e3 + 3)+..........+(en+ 1),where e repreents the exponent of prime factors respectively. Now a we are given that unique prime factors are 12 it means all the factors are different. So 12=(e1+ 1)(e2+1)+....+(en+1),which is possible only as below:
3*2*2=(2+1)(1+1)(1+1) Hence it implies there can be maximum 3 prime factors whose exponents are 2,1,1 respectively. Therefore B IMO.



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 10:50
If N has prime factorisation as a^p * b^q * c^r, where a,b,c are prime numbers. Then the number of factors of N are (p+1)(q+1)(r+1) and so on.
Here in the question we need to maximise prime factors of N. This can be done by factorising 12 as 2*2*3 Prime factorisation of N now be represented as a^1*b^1*c^2
So can have maximum of 3 prime numbers.
B is correct. The number of factors of N can be calculated with the help of



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 10:51
Exactly 12 unique factors,so as per the formula for factors it's (n+1)*(m+1)*....where n,m are power of the prime factors So 12=3*4 or 2*3*2 In case of 2*3*2 it will be max no of prime factors like we can say example of 1such number to be 2*3*3*5=60
So max no of unique prime factors will be 3 so 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, 11:20
Given,
1. N has exactly 12 unique factors.
To find,
Largest possible number of unique prime factors that N could have.
Formula to get number of factors of a number x is,
x = a^m * b^n * c*o where a, b, c are unique prime factors of x
=> Number of factors of x = (m + 1) * (n + 1) * (o + 1)
12 can be expressed in 4 ways 
1. 1 * 12
2. 2 * 2 * 3
3. 4 * 3
4. 2 * 6
Here, each of these numbers  1 would represent the power of distinct prime factors of x.
Hence as per these representations of 12, 3 is the largest possible number of unique prime factors that N could have.
Answer: B




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