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

Question

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

geezus24x7 · Accepted Answer

N has 12 factors so it isn't a perfect square.

Total number of factors of N= a^m * b^n *... is calculated as-
(m+1)*(n+1)*..
To get 12 as total factors, largest possible number of unique prime factors is 3-
a^1*b^1*c^2

=> (1+1)*(1+1)*(2+1) = 12

Hence option B

Sayon · Answer

Positive integer N can have exactly 12 unique factors in any of the following ways:

12*1 -> (11 + 1) ->   A^11   ->  1 prime factor 
6*2 -> (5+1)(1+1) ->   A^5*B^1   ->  2 prime factors 
4*3 -> (3+1)(2+1) ->   A^3*B^2   ->  2 prime factors 
2*2*3 -> (1+1)(1+1)(2+1) ->  A^1*B^1*C^2   ->  3 prime factors  –   This one being the largest.

Answer B

Kinshook · Answer

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

No of unique factors of positive integer N = 12
12 = 2*2*3 = 4*3 = 2*6 = 12
Therefore
 
 N = P_1*P_2*P_3^2  for 12 = 2*2*3. No of unique prime factors = 3 (1)
or
 N = P_1^3*P_2^2  for 12 = 4*3 No of unique prime factors = 2 (2)
or
 N= P_1*P_2^5  for 12 = 2*6 No of unique prime factors = 2 (3)
or
 N= P_1^11  for 12 = 12 No of unique prime factors = 1 (4)

We see that for equation (1), no of unique prime factors = 3 is largest possible number of unique prime factors

IMO B

nick1816 · Answer

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

duchessjs · Answer

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} 	o \mathtt{12 unique factors 
OR
 \mathtt{(7^1)(5^2)(11^1) = 1925} 	o \mathtt{12 unique factors 
OR
 \mathtt{(5^1)(3^2)(2^1) = 90} 	o \mathtt{12 unique factors

The following technique is used to determine prime factors and factor number totals...

\

Correct Answer: B. 3

JonShukhrat · Answer

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

RastogiSarthak99 · Answer

Seems troubling at first to look at but fairly easy-ish. Think about it.

No. of unique factors is determined by the power of prime numbers for N in this case. Ignore the prime numbers, because we only truly care about the exponent.

Formula for calculating unique factors if 3^x * 5^y then unique factors = (x+1) (y+1) (Ignore the fact that I've taken 3 and 5, just for representation)

Now, to reach 12 there are the following ways:
1 x 12
2 x 6 
3 x 4

The question asks us for largest number of unique prime factors.

2x 6 can be written as 2 x 3 x 2 (Suggesting that --> (1+1) * (2+1) * (2+1) meaning 3 unique prime factors. Same for 3 x 4 (3 x 2 x 2)

I chose B this way.

KarishmaB · Answer

We know how to obtain the total number of factors of N. If N = a^x * b^y * c^z ... Total number of factors = (x + 1)(y+1)(z+1)... given a, b and c are distinct prime factors. Prime factorise 12 to get 2*2*3 (we cannot split 12 into smaller factors) So N can have 3 distinct prime factors e.g. N = 5*7*11^2 Total number of factor here = (1 + 1)(1+1)(2+1) = 12 Answer (B) Check this video on factorisation: https://youtu.be/Kd-4cH4cqHw

s98vcrd2d · Answer

how the heck is 2x2 two seperate distinct prime factors

Bunuel · Answer

Which solution above claims that?

N having 12 factors implies that assuming  N = p^a * q^b * r^c *... ,  (a + 1)(b + 1)(c + 1)... = 12 .

The maximum number of ways 12 can be broken into the product of multiples greater than 1 is 3: 12 = (1 + 1)(1 + 1)(2 + 1). Thus, the largest possible number of unique prime factors that N could have is 3. For example, N could be 2 * 3 * 5^2, which has (1 + 1)(1 + 1)(2 + 1) = 12 positive factors and three unique prime factors: 2, 3, and 5.

Answer: B.

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

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