rishab0507 wrote:
Kinshook wrote:
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
Hello ,
I have small doubt as i am new to this concept. I understand rule of finding factors : power +1 : But question stem says N has 12 unique factors. In all cases you presented, if P2^2 or P^3 it means, its p2*p2 or p3*p3*p3, How can we take this as question says it has Unique factors, but from this we have 2 times P2 and 3 times P3, which cannot be unique as its repeating. Don't condition itself falls at time of assumption as it doesnot have unique 12 factors, Can't it be p1,p2,p3----p12 .
Hope i am able to explain my question
When we say N = a^p * b^q, then we mean that N has (p + 1) * (q + 1) unique factors.
For example 72 = 2^3 * 3^2
Hence, 72 has (3 + 1) * (2 + 1) = 12 unique factors.
The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
As you can see, there are 12 unique factors and nothing is repeated.
Hope this helps.
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