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Re: Positive integer N has exactly 12 unique factors. What is the largest
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18 Jul 2019, 22:17
Question: Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?Unique factors of a number can be calculated by finding all prime factors, adding 1 to each unique prime factor quantity, then multiplying the resulting numbers. The number 12 can be broken down to (2)(3)(2), this break down means we need to find a value for N with two prime factors with a quantity of one each, and one prime factor with a quantity of two. Any three prime factors will do. For example... \(\mathtt{(3^1)(2^2)(7^1) = 84} \to \mathtt{12 unique factors\) OR \(\mathtt{(7^1)(5^2)(11^1) = 1925} \to \mathtt{12 unique factors\) OR \(\mathtt{(5^1)(3^2)(2^1) = 90} \to \mathtt{12 unique factors\) The following technique is used to determine prime factors and factor number totals... \[\text{Prime Factorization Calculation }\\ \text{Example: N = 90} \\ \begin{pmatrix*} &&90&& \\ &\swarrow&&\searrow&\\ 2&&\downarrow&&5\\ &&9&&\\ &\swarrow&&\searrow&\\ 3&&&&3 \end{pmatrix*} \rightarrow \text{Prime Factors if N are } 2,3,3,5\\ \] \[\text{Unique Factor Calculation }\\ \text{Example: N = 90}\\ \begin{pmatrix*} Prime\text{ Factor } &\text{ Add 1 }& \text{ # of Factors } \\ 2^1 & 2^{1+1} & 2 \\ 3^2 & 3^{2+1} & 3 \\ 5^1 & 5^{1+1} & 2\\ \end{pmatrix*} \rightarrow\text{ Total number of factors are (2)(3)(2) = 12}\\ \\ \text{Prime Factors of N are } 2,3,3,5\\ \text{Unique factors remove duplicates, leaving } 2,3,5 = \text{ 3 Total Unique Factors} \] Correct Answer: B. 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, 22:26
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
Possible factorization of N = x^3*y^2, x^5*y, x^11
here, largest possible number of unique prime factors should be 11.
So, the correct answer choice is (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, 22:31
Formula: If N = \(2^a * 3^b * 5^c * 7^d *\) . . . . . . Number of positive factors of N = (a + 1)*(b + 1)*(c + 1)*(d + 1)* . . . . . . .
12 = 2*2*3 = (1 + 1)*(1 + 1)*(2 + 1) So, Possible values of N = 2*3*\(5^2\) or 3*5*\(7^2\) or so on ......
Note that all above possible values have ONLY 3 unique prime factors
IMO 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, 22:34
Answer D 12 unique factor can be 11 prime numbers and one digit 1. So maximum prime factor can be 11 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, 22:57
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have? Out of the 12 unique factors, one has to be the number itself. Remaining 11 can all be prime numbers. Largest number of unique prime factors = 11. Option 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, 23:06
We know that if \(N\) is prime factorized, it can be written as \(a^x\) or as \(a^x*b^y\) or as \(a^x*b^y*c^z\) depending on the number of prime factors. \(a, b,\) and \(c\)  prime factors of \(N\) \(x, y,\) and \(z\)  powers of these prime factors of \(N\) 1. The least possible number of unique prime factors of \(N\) is \(1\) when \(N\) is \(11th\) power of a prime number: If \(N=a^x\) and \(x=11\), then the unique factors of \(a^1\) is \((11+1)=12\) 2. \(N\) can have \(2\) unique prime factors: If \(N=a^x*b^y\) and \(x=2, y=3,\) then unique factors of \(a^2*b^3\) is \((2+1)(3+1)=12\) If \(N=a^x*b^y\) and \(x=5, y=1,\) then unique factors of \(a^5*b^1\) is \((5+1)(1+1)=12\) 3. The largest possible number of unique prime factors of \(N\) is \(3\): If \(N=a^x*b^y*c^z\) and \(x=1, y=1, z=2\) then unique factors of \(a^1*b^1*c^2\) is \((1+1)(1+1)(2+1)=12\) Powers can be in any order: \(N=a^1*b^1*c^2\) \(N=a^1*b^2*c^1\) \(N=a^2*b^1*c^1\) In any case \(N\) has at most \(3\) unique prime factors provided that \(N\) has exactly \(12\) unique factors. Hence 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, 23:10
The maximum number of unique prime factors can be equal to number of unique factors ,,ie 12 . But 1 will also be a factor , and Is not prime. So the answer will be 121 or 11 or OPTION 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, 23:36
There is an important formula to find the factors of a number 'N'
if \(N=(a^m)*(b^n)*(c^o)*(D^p)\)... (where a,b,c,.. are prime factors of N) Number of all factors of \(N = (m+1)*(n+1)*(o+1)*(p+1)\)....
Total number of unique factors = 12= 2 * 2 * 3 = (1+1)(1+1)(2+1)
\(N=(a^1)*(b^1)*(c^2)\)
So largest possible number of unique prime number has to 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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19 Jul 2019, 00:45
Prime factorization of 12=2*2*3 > This is the product of powers of factors of N (Each power incremented by 1) That is, N=\(X^{1} * Y^{1}*Z^{2}\) Therefore number of unique prime factors is 3. Ans: B
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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19 Jul 2019, 00:56
Positive integer N has exactly 12 unique factors
N could be = \(a^3\)*\(b^2\) (where a and b are prime factors ) total factors = (3+1)(2+1) = 12
N could be = \(a^1\)*\(b^1\)*\(c^2\) (where a ,b , and c are prime factors) total unique factors = (1+1)(1+1)(2+1) = 12
when there are 4 prime factors N = \(a^1\)*\(b^1\)*\(c^1\)*\(d^1\) then N will have = 2*2*2*2 = 16 factors so 4 prime factors are not possible
so for exactly 12 unique factors largest possible number of unique prime factors are = 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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19 Jul 2019, 01:21
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
N exactly having 12 unique factors.
Lets say n consists of 4 prime factor , then how many combinations of multiples of prime and unique factor I can make = 4c1+ 4c2 + 4c3+4c4 =15 and 1 is factor for all so total = 16
lets say3 prime factor , n =3 , then Total unique factors = 3c1+3c2+3c3 = 3+3+1 = 7 , add 1 to it = 8
so n is between , 4<n<3 , this is possible when n=3 but a prime factor repeats lets say N has = 2,3,5 and 2 repeats { 2,2,3,5} then no . of unique factors = 3(2;3;5)+4( 2,2 ; 2,3; 2,5; 3,5) +3 (2,3,5);(2,2,3);(2,2,5) + 1 (2,2,3,5) = 11 And 1 as a factor for N , Total factor = 12
So total prime unique factors possible = 3  Option B



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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19 Jul 2019, 03:02
Number of factors of N = 12 Let prime factors of N be \(p_1,p_2,p_3...\) 12 can be factorized as follows: 1x12 > Possible value of N = \(p_1^{11}\) 2x2x3 > Possible value of N = \(p_1^1*p_2^1*p_3^2\)
As \(p_1,p_2,p_3\) are unique, maximum no. of prime factors = 3
Option (B).



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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19 Jul 2019, 03:04
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: First make prime factorization of an integer n=\(a^p\) X \(b^q\) X \(c^r\), where a, b, and c are prime factors of n and p, q, and r are their powers. The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). NOTE: this will include 1 and n itself. When we consider N, Number of unique factors of N = \(a^p\) X\(b^q\) X\(c^r\) we know that 12 is derived from multiplication of 3 different exponents maximum, i.e 2 X 2 X 3, Hence we can conclude that these are the exponents to 3 unique prime numbers So the answer must be B
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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19 Jul 2019, 03:44
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 can have either 2 or 3 unique prime factors. But since we need largest number, answer is B, 3 factors. Consider 150=2^1*3^1*5^2, has 12 unique factors, but only 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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19 Jul 2019, 04:34
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
N = a^k0*b^k1*c^k2... a,b,c  prime factors #ofFactors = (k0+1)*(k1+1)*(k2+1) If N has exactly 12 unique factors: #ofFactorsMax = 2*2*3 That's mean the largest possible number of unique prime factors is 3
ANSWER B
(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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19 Jul 2019, 05:29
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
The formula of finding the number of factors of certain number: > (a1+1)(a2+1)(a3+1)....=12 (a1,a2,a3..the power of Prime numbers)
The optimum condition of finding the number of unique factors of N is (1+1)(1+1)(2+1)=12 > N could have maximum 3 Unique Prime Factors (minimum 1 uni.pri.factors)
The answer choice is B.



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Positive integer N has exactly 12 unique factors. What is the largest
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19 Jul 2019, 05:46
Positive integer N has exactly 12 unique factors. What is the largest possible number of unique prime factors that N could have?
If a, b,c,d... are prime factors of X Formula to derive number of unique factors of \(X = (a1+1)*(b1+1)*(c1+1)*(d1+1)*\)...., where a1, b1,c1, d1 ... are corresponding powers of a,b,c,d in X we know that \((a1+1)*(b1+1)*(c1+1)*(d1+1)*.... = 12\) To get max number of prime factors, a1,b1.... must be minimum >0 and therefore =1 Now,\((1+1)(1+1)*(1+1)*(1+1)\) already > 12 (namely we get 16), therefore, maximum number of unique prime factors of N is 3
Generalizing, we should set up equation such as \(2^{n}<= (number_of_unique_factors)\) to get max number of unique prime factors of N.
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, 06:26
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
Unique factors 12 means (a+1)(b+1)...= 12
(11+1) or 2^11 so unique pf is 1 (5+1)(1+1) unique PF is 2 (3+1)(2+1) unique PF is 2 (1+1) (1+1)(2+1) unique PF is 3
So max is B.3
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Re: Positive integer N has exactly 12 unique factors. What is the largest
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19 Jul 2019, 06:03
The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). 12 can be obtained in following ways (11+1) (1+1)(5+1) (2+1)(3+1); (1+1)(1+1)(2+1)=12 Maximum three factors, answer B



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Re: Positive integer N has exactly 12 unique factors. What is the largest
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19 Jul 2019, 07:59
12 factors means 2x2x3 if we consider the lowest factors of 12.
To find out the total number of factors of a number, say 15, we express it as the multiplication of the prime factors, add 1 to each exponent and then multiply the increased exponents. Therefore for 2,2, and 3 we have a total of 3 unique factors.




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