I understand this concept but just for in depth clarity please claarify that the above method will list all the factor including negatives and positives ? so I
divided 32/2 to get 16 positive and 16 negative factors ...please clarify?

In the context of GRE,we stick to factoring positive integers to sidestep negative factor issue. We deal with it in higher mathematics though!

If the number is positive only condsider positive factors. I doubt you will find negative numbers in the factorization problems.

Hwo do we get to 16,000= (2^7)(5^3) Is it trial and error?

Final goal is to reach 1.

When asking for prime factors we start with the smallest prime factor 2.

We can clearly see that in 16000 we would have many multiples of 2. So one way is to keep dividing by 2 till we can divide no more.

16000/2 = 8000 ...... one factor of 2
8000/2 = 4000 ..........second factor 2
.
.
1000/2= 500 .............. 5th factor factor of 2
.
250/2=125 ................. 7th Factor of 2

Now we need other prime factors so we move on to higher prime number such as 3 and 5.

Division by 3 is not possible so 5

125 can be divided 3 times with 5 to get 1.

Hence \(16000=2^7 \times 5^3\)

Here's a video explaining how to find the prime factorization of a number:

Is there a way to factor out a big number like this faster on the test, or do you recommend we just go for the smallest prime numbers?

Ideally you can break the number into smaller number whose factors you know such as

\(16000 = 16 \times 1000\) Now 1000 is \(10^3\) or \(2^3 \times 5^3\) and 16 is \(2^4\)

So 16000 is \(2^4 \times 2^3 \times 5^3\).

The number 16,000 has how many positive divisors?

----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-------------------------

16,000 = (2)(2)(2)(2)(2)(2)(2)(5)(5)(5)
= (2^7)(5^3)
So, the number of positive divisors of 16,000 = (7+1)(3+1)
= (8)(4)
= 32

Answer: 32

RELATED VIDEO FROM OUR COURSE

Is there a way to solve this question that doesn't involve remembering the 'N has a total of (a+1)(b+1)(c+1)(etc) positive divisors'. Is there a method of solving that prioritizes logic over formula memorization?

You can use logic to answer this question.
In the following video, I explain how the formula is derived.
You can use the same logic to answer the above question

16000
=16×10^3
=2^4×(2×5)^3
=2^7×5^3
if 2^n×5^m ,then total positive factors or divisors will be (n+1)(m+1)



The rules is to get total divisor or total positive factors ,you have to add+1 with power of each prime factors and multiply
(7+1)×(3+1)
=8×4
=32



You will encounter this kind of math lot throughout your your whole GRE preparation, better spend some times and watch a video in YouTube about how to find total factors

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