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

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

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New post 18 Jul 2019, 22:17
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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  [#permalink]

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New post 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  [#permalink]

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New post 18 Jul 2019, 22:31
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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  [#permalink]

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New post 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

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

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New post 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  [#permalink]

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New post 18 Jul 2019, 23:06
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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  [#permalink]

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New post 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 12-1 or 11 or OPTION D


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

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New post 18 Jul 2019, 23:36
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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  [#permalink]

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New post 19 Jul 2019, 00:45
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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  [#permalink]

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New post 19 Jul 2019, 00:56
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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  [#permalink]

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New post 19 Jul 2019, 01:21
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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?

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  [#permalink]

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New post 19 Jul 2019, 03:02
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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  [#permalink]

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New post 19 Jul 2019, 03:04
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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

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  [#permalink]

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New post 19 Jul 2019, 03:44
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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?
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  [#permalink]

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New post 19 Jul 2019, 04:34
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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?

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  [#permalink]

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New post 19 Jul 2019, 05:29
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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?

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  [#permalink]

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New post 19 Jul 2019, 05:46
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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?

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  [#permalink]

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New post 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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Originally posted by shruthiarvindh on 19 Jul 2019, 05:50.
Last edited by shruthiarvindh on 19 Jul 2019, 06:26, 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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New post 19 Jul 2019, 06:03
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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  [#permalink]

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New post 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.
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Re: Positive integer N has exactly 12 unique factors. What is the largest   [#permalink] 19 Jul 2019, 07:59

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