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2,600 has how many positive divisors?

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by DavidG@VeritasPrep » Wed Nov 08, 2017 8:21 am
VJesus12 wrote:2,600 has how many positive divisors?

A. 6
B. 12
C. 18
D. 24
E. 48

The OA is D.

How can I calculate the total of divisors? Is there any formula? Experts, may you help me?
Neat shortcut for finding the number of divisors of a given value.

First, take the prime factorization of the number in question: 2600 = 26 * 100 = 2 * 13 * 2^2 * 5^2 = 2^3 * 5^2 * 13

Next, add one to the exponent of each prime base and multiply the results. (Note that 13 is the same as 13^1): (3+1)(2+1)(1+1) = (4)(3)(2) = 24. The answer is D
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by EconomistGMATTutor » Wed Nov 08, 2017 11:14 am
2,600 has how many positive divisors?

A. 6
B. 12
C. 18
D. 24
E. 48

The OA is D.

How can I calculate the total of divisors? Is there any formula? Experts, may you help me?
Hi VJesus12,
Let's take a look at your question.

We are asked to find the number of positive divisors of 2600.
In order to find number of positive divisors a number has, we find the prime factorization, and add one to all the exponents and multiply them.

Let's write 2600 as a product of its prime factors.
$$2600=\ 26\ \times100$$
$$2600=\ 2\times13\ \times10\times10$$
$$2600=\ 2\times13\ \times2\times5\times2\times5$$

2600 is now written as a product of its prime factors.
Let's now write the factors using exponents.
$$2600=\ 2^3\times5^2\times13^1$$

To find the number of positive divisors of 2600, add 1 to each exponent and multiply them.
$$=\left(3+1\right)\left(2+1\right)\left(1+1\right)$$
$$=\left(4\right)\left(3\right)\left(2\right)$$
$$=24$$

Therefore, Option D is correct.

Hope it helps.
I am available if you'd like any follow up.
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by Brent@GMATPrepNow » Fri Nov 10, 2017 8:53 am
VJesus12 wrote:2,600 has how many positive divisors?

A. 6
B. 12
C. 18
D. 24
E. 48
--------ASIDE------------------------------
IMPORTANT: 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---------------------

2,600 has how many positive divisors?
2600 = (2^3)(5^2)(13^1)
So, the number of positive divisors of 2600 = (3+1)(2+1)(1+1)
=(4)(3)(2)
= 24
= D

Cheers,
Brent
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by Matt@VeritasPrep » Fri Nov 10, 2017 1:22 pm
If it's easier to see this visually, check out this great tutorial.

In the future, remember that Google is your friend! In a pinch you can usually find walkthroughs of most basic formulas such as this without having to wait a few hours for an expert to reply.
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2,600 has how many positive divisors?

by Brent@GMATPrepNow » Fri Nov 10, 2017 2:53 pm
Here's a video that explains the rule AND explains why it works: https://www.gmatprepnow.com/module/gmat ... /video/828

Cheers,
Brent
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