How many positive integers have more than 3 positive divisor

Question

How many positive integers have more than 3 positive divisors and are factors of 1,800?

(A) 24
(B) 26
(C) 27
(D) 29
(E) 30

percyboi66 · Answer

I got the correct answer (D) but had to take a very roundabout way by basically writing down all the factors of 1800, is there a quicker way?

kevincan · Answer

If  p  is a prime number, how many divisors does  p^2  have?

kevincan · Answer

From the total number of factors, subtract the number that have at most three positive divisors.

Adit_ · Answer

Use the formula for total number of factors of a number by prime factorizing.

1800 = 2^3*3^2*5^2.
Total factors = 4*3*3 = 36.

Now you need to remove those factors whose number of factors of those factors are 3 or less than 3. 
When does that happen?
When you have a^2 form or just a form. Every other face you will have more than 3 factors for that factor.
Also dont forget about 1.

Now from prime factorization of 1800 we can remove the single factors ie- 2,3,5 which is 3.
We should also remove a^2 form which are 2^2,3^2,5^2 which is another 3.
Including 1 it is 3+3+1 = 7 factors removed.

Final answer = 36-7 =  29 .

pappal · Answer

total factors of (1800=2^3*3^2*5^2)=36
out of these ones which don't have more than 3 factors are,1,2,3,4,5,9 & 25
hence required ones=36-7=29
D

Ravisoni · Answer

Sol:

As, 1800=2^3 x 3^2 x5^2 
Total factors= 4x3x3 = 36
No. of Prime factor(i.e 2,3,5) = 3 (having 3 positive divisors)
No. of Prime factor(i.e 4,9,25) = 3 (having 3 positive divisors)
Ans = Total factors - prime factors-square of prime factor - 1(1 having only 1 divisor)
 = 36 - 3- 3 - 1 = 29 (D)

How many positive integers have more than 3 positive divisor

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How many positive integers have more than 3 positive divisor

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