If m, p, and t are Distinct Positive Prime Numbers, then (m^3)(p)(t) has How Many Different Positive Factors Greater than 1?

Question: If m, p, and t are distinct positive prime numbers, then (m^3)(p)(t) has how many different positive factors greater than 1?

1. 8
2. 9
3. 12
4. 15
5. 27

“If m, p, and t are distinct positive prime numbers, then (m^3)(p)(t) has how many different positive factors greater than 1?” – is the topic of GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

For example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40

Now, let us come to the given problem statement:

(m^3)(p)(t) = (m^3)(p^1)(t^1)
Hence, the positive divisors of (m^3)(p)(t) = (3+1)(1+1)(1+1) = (4)(2)(2) = 16

We have included 1 as one of the 16 factors in our solution above. The question asks us to find the number of positive factors greater than 1. So we will have to exclude 1 from it.

So, the answer to the question = 16 - 1 = 15

Hence, the correct answer is D.

Approach Solution 2:

Let Number is
(m^3)(p)(t)=(2^3)(3)(5)=120

We can write 120 as product of two numbers in following ways
1*120
2*60
3*40
4*30
5*24
6*20
8*15
10*12

So we get 8 cases in total= 8*2 i.e. 16 factors (including 1)
Since we included 1 but the problem statement asks greater than 1. So we will exclude 1 from the solution.
We get:
Factors greater than 1 = 16-1=15

Hence, D is the correct answer.

Approach Solution 3:

We need to find the number of factors of (2^a)*(3^b)*(5^c) ... = (a+1)(b+1)(c+1) ...

Let us consider m, p and t are the distinct prime numbers.
These number are already represented in its prime factorization form
Number of factors = (3+1)(1+1)(1+1) = 16
Out of these, one factor would be 1.

Hence different positive factors greater than 1 = 15

Hence, D is the correct answer.

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