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

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


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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

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

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

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