If m, p, and t are distinct positive prime numbers, then (m^3)(p)(t)

Number of factors of (2^a)*(3^b)*(5^c) ... = (a+1)(b+1)(c+1) ...

If m, p and t are the distinct prime numbers, then the number is 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
Correct Option: D

Method 1:

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

8 cases = 8*2 i.e. 16 factors (including 1)

Factors greater than 1 = 15

Method 2:



So factors of \((m^3)(p)(t)\) will be \((3+1)(1+1)(1+1) = 16\) but these factors include 1 as well

therefore factors of \((m^3)(p)(t)\) greater than 1 = 16-1 = 15


Answer: Option D

Okay what does "positive " prime numbers signify here ?
Can primes be negative too ?
NEVER EVER EVER.

Regards
StoneCold
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I wonder that 16 is not among the answers))

You need factors greater than 1. So the number of factors is 16 - 1 = 15.
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Yes i understand this. But GMAT likes to use traps so one can forget to exclude 1 and choose 16. But this answer is not present

I don't get this one. Is it a formula that you add +1 to each exponent to get # of factors?

Check out this video: https://youtu.be/Kd-4cH4cqHw
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To determine the total number of factors of a number, we can add 1 to the exponent of each distinct prime number and multiply together the resulting numbers.

Thus, (m^3)(p)(t) = (m^3)(p^1)(t^1) has (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16 total factors. Since 1 is one of those 16 factors, there are actually 15 different positive factors greater than 1.

Answer: D
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total number of factors = (3+1)(1+1)(1+1) = 16
except 1 number of factors = 16-1 = 15

m^3*p*t

Total no. of factors= (3+1)(1+1)(1+1)= 4*2*2=16

But 1 is excluded. Therefore the answer is 16-1=15

----ASIDE---------------------

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.

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

-----ONTO THE QUESTION!!----------------------------

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

IMPORTANT: We have included 1 as one of the 16 factors in our solution above, but the question asks us to find the number of positive factors greater than 1.
So, the answer to the question = 16 - 1 = 15

Answer: D

RELATED VIDEO

From the GMAT Official Guide:
A prime number is a positive integer that has exactly two different positive divisors, 1 and itself.
For example, 2, 3, 5, 7, 11, and 13 are prime numbers, but 15 is not, since 15 has four different positive divisors, 1, 3, 5, and 15.
The number 1 is not a prime number since it has only one positive divisor.
Every integer greater than 1 either is prime or can be uniquely expressed as a product of prime factors. For example, 14 = (2)(7), 81 = (3)(3)(3)(3), and 484 = (2)(2)(11)(11).

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

Total number of positive factors of (m^3)(p)(t) = 4*2*2 = 16
Number of different positive factors greater than 1 = 16- 1 = 15

IMO D

How do I make sure that with the formula of adding +1 to the exponent, I do not double count any factors? So also that 1 is counted only once

That's how the formula works.

Let's say the prime factorization of the number is \(p^a * q^b * r^c..\). where p, q, r are prime numbers and a, b, c are their respective exponents. The total number of factors of the given number is then \((a+1)(b+1)(c+1)...\). This will include 1 and the number itself.

Why? Because each factor includes zero or more instances of each prime number. If p is a factor, it might be there 0 times, 1 time, 2 times, all the way up to a times. Hence, there are (a+1) possibilities for p, and similarly for q, r, and so on. Since these possibilities for each factor are independent, you multiply the possibilities to find the total.

For example, if a number is 36, its prime factors are 2 and 3. In terms of prime factorization, 36 can be written as 2^2 * 3^2. So, the total number of factors will be (2+1)*(2+1) = 9. These factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.

I hope this makes it easier to understand.
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