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

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

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Updated on: 18 Jul 2019, 19:45
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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?

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  [#permalink]

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18 Jul 2019, 08:56
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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 $$c=3-1=2$$.

Hence, the largest number of prime factors for number $$N$$ is $$3$$: $$A^1*B^1*C^2$$.

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

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18 Jul 2019, 08:57
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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  [#permalink]

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18 Jul 2019, 08:57
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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 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.)

Please hit kudos if you like the solution.
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Re: Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 08:57
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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 3

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

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18 Jul 2019, 09:03
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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  [#permalink]

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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

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

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18 Jul 2019, 09:05
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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?

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  [#permalink]

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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  [#permalink]

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18 Jul 2019, 09:07
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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  [#permalink]

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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  [#permalink]

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Updated on: 19 Jul 2019, 01:13
1
IMO-B

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  [#permalink]

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18 Jul 2019, 09:25
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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 non-negative 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  [#permalink]

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18 Jul 2019, 09:27
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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?

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 $$x^{(3-1)}*y^{(2-1)}*z^{(2-1)}$$... where x, y, and z are unique prime factors

ANSWER: B-3 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  [#permalink]

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18 Jul 2019, 09:29
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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.

Regards,
Rishav
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Positive integer N has exactly 12 unique factors. What is the largest  [#permalink]

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18 Jul 2019, 09:59
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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 = 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

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

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18 Jul 2019, 10:33
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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:
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  [#permalink]

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18 Jul 2019, 10:50
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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  [#permalink]

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18 Jul 2019, 10:51
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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  [#permalink]

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18 Jul 2019, 11:20
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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.

Re: Positive integer N has exactly 12 unique factors. What is the largest   [#permalink] 18 Jul 2019, 11:20

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