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# Which of the following has as many positive factors as 300 have? A) 20

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Which of the following has as many positive factors as 300 have? A) 20  [#permalink]

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09 Jun 2017, 04:19
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67% (01:41) correct 33% (01:07) wrong based on 150 sessions

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Which of the following has as many positive factors as 300 have?
A) 200
B) 400
C) 500
D) 600
E) 700

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Which of the following has as many positive factors as 300 have? A) 20  [#permalink]

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09 Jun 2017, 04:30
300, when prime factorized gives $$2^2 * 3 * 5^2$$ which has 3*2*3 = 18 factors.

Of the lot only 700 has 3 distinct prime factors and the same power of factors($$2^2 * 5^2 *7$$)
Hence, Option E is our answer.
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Re: Which of the following has as many positive factors as 300 have? A) 20  [#permalink]

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09 Jun 2017, 08:07
Imo E
13 has 18 factors
This only 700 has 18 factors

Sent from my ONE E1003 using GMAT Club Forum mobile app
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Re: Which of the following has as many positive factors as 300 have? A) 20  [#permalink]

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09 Jun 2017, 08:29
Top Contributor
GMATinsight wrote:
Which of the following has as many positive factors as 300 have?
A) 200
B) 400
C) 500
D) 600
E) 700

---------ASIDE---------------------------------
If N = (p^a)(q^b)(r^c)..., where p, q, r,...(etc.) are prime numbers, then the total number of positive divisors of N is equal to (a+1)(b+1)(c+1)...

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 ONTO THE QUESTION--------------------------
300 = (2^2)(3^1)(5^2)
So, the number of positive divisors of 300 = (2+1)(1+1)(2+1) = (3)(2)(3) = 18

NOTE: this is one of those questions that require us to check/test each answer choice. In these situations, always check the answer choices from E to A, because the correct answer is typically closer to the bottom than to the top.
For more on this strategy, see my article: http://www.gmatprepnow.com/articles/han ... -questions

E) 700
700 = (2^2)(5^2)(7^1)
So, the number of positive divisors of 700 = (2+1)(2+1)(1+1) = (3)(3)(2) = 18
We have a MATCH!!

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Re: Which of the following has as many positive factors as 300 have? A) 20  [#permalink]

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09 Jun 2017, 11:15
300 = 3*100, and here '3' has nothing in common (no common factor) with 100 (which is made up of 2 and 5).

If we scan the answer choices,

200 = 2*100.. but 2 has a common factor with 100
400 = 4*100.. but 4 has common factors with 100
500 = 5*100.. but 5 has a common factor with 100
600 = 6*100.. but 6 has common factors with 100
700 = 7*100. Bingo! 7 Does NOT have any common factor with 100. This is our answer

We could apply this approach here, and not use time in writing prime factorisation of each number - just because all the answer choices are similar to each other (200, 400, 500, 600, 700) and also to the one given in the question stem (300).

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Re: Which of the following has as many positive factors as 300 have? A) 20  [#permalink]

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13 Jun 2017, 12:10
GMATinsight wrote:
Which of the following has as many positive factors as 300 have?
A) 200
B) 400
C) 500
D) 600
E) 700

We may recall that the number of total factors of a number can be found by breaking the number into prime factors and adding 1 to each exponent from each unique prime factor. Let’s start with 300:

300 = 30 x 10 = 2 x 3 x 5 x 2 x 5 = 2^2 x 3^1 x 5^2

Thus, 300 has (2+1)(1+1)(2+1) = 3 x 2 x 3 = 18 total factors.

Let’s now analyze each answer choice:

A) 200

200 = 20 x 10 = 2 x 2 x 5 x 2 x 5 = 2^3 x 5^2

200 has (3 + 1)(2 + 1) = 4 x 3 = 12 total factors.

Answer choice A is not correct.

B) 400

400 = 40 x 10 = 2 x 2 x 2 x 5 x 2 x 5 = 2^4 x 5^2

400 has (4 + 1)(2 + 1) = 5 x 3 = 15 total factors.

Answer choice B is not correct.

C) 500

500 = 50 x 10 = 5 x 2 x 5 x 2 x 5 = 2^2 x 5^3

500 has (2 + 1)(3 + 1) = 3 x 4 = 12 total factors.

Answer choice C is not correct.

D) 600

600 = 60 x 10 = 2 x 3 x 2 x 5 x 2 x 5 = 2^3 x 3^1 x 5^2

600 has (3 + 1)(1 + 1)(2 + 1) = 4 x 2 x 3 = 24 total factors.

Answer choice D is not correct.

E) 700

700 = 70 x 10 = 7 x 2 x 5 x 2 x 5 = 2^2 x 5^2 x 7^1

700 has (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18 total factors.

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Re: Which of the following has as many positive factors as 300 have? A) 20  [#permalink]

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02 Jan 2019, 22:41
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Re: Which of the following has as many positive factors as 300 have? A) 20   [#permalink] 02 Jan 2019, 22:41
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