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

Question

Which of the following has as many positive factors as 300 have? A) 200 B) 400 C) 500 D) 600 E) 700 Source: www.GMATinsight.com

BrentGMATPrepNow · Accepted Answer

---------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 Now check the answer choices.... 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: https://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!! Answer: E RELATED VIDEO https://www.youtube.com/watch?v=lVZ9GXd6ZBE

pushpitkc · Answer

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.

arvind910619 · Answer

Imo E 
13 has 18 factors 
This only 700 has 18 factors

Sent from my ONE E1003 using  GMAT Club Forum mobile app

amanvermagmat · Answer

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).

Hence  E answer

JeffTargetTestPrep · Answer

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.
 
Answer choice E is correct.
 
Answer: E

TheNightKing · Answer

300 = 3*100 = 3^1 * 2^2 * 5^2 = 2*3*3 = 18

We are looking for Prime number (that is not a factor of 100) multiplied by 100.

A) 200 - No. 2 is a factor of 100.
B) 400 - No
C) 500 - No
D) 600 - No. 6 is not prime.
E) 700 - 7 * 100 Yes

Answer E

Anuramaa · Answer

300 has 3^1 * 5^2 * 5^2 = (1+1)+(2+1)+(2+1) = 18 factors

look for another prime number in hundreds place which is not 5 or 2

700 has 7^1 * 5^2 * 5^2 = (1+1) + (2+1) + (2+1) = 18 factors

looking for prime other than 5 or 2 because.

500 will make it 5^3 * 2^2 = (3+1) + (2+1) = 12 factors

same logic for not choosing 2 also

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

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