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How many integers from 1 to 900 inclusive have exactly 3 positive divi

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Bunuel
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How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   10 Jul 2019, 01:32
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How many integers from 1 to 900 inclusive have exactly 3 positive divisors?

(A) 10
(B) 14
(C) 15
(D) 29
(E) 30
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A
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   10 Jul 2019, 01:50
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This is prime numbers problem.
First, for having exactly 3 divisors, the number should be in the form of X = \(a^2\)
Then only the number of divisors will be 3.
Now, The prime number is divisible by itself and 1. So, the squares of those prime numbers will also divisible by 1, square of respective prime number and itself, thus having 3 divisors.
So, numbers are => 4,9,25,49,121,169,289,361,529,841
These are the squares of prime numbers of : 2,3,5,7,11,13,17,19,23,29

Thus total of 10 numbers.

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How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   10 Jul 2019, 01:36
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A number with exactly 3 positive factors must be the square of a prime

So, the numbers satisfying this condition must be squares of 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 - 10 numbers

Any square number after this exceeds 900

Answer is (A)
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   10 Jul 2019, 02:38
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Bunuel wrote:
How many integers from 1 to 900 inclusive have exactly 3 positive divisors?

(A) 10
(B) 14
(C) 15
(D) 29
(E) 30


square of prime no will have exactly 3 +ve factors ; 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
IMO A: 10
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   10 Jul 2019, 02:44
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IMO : A
How many integers from 1 to 900 inclusive have exactly 3 positive divisors?

(A) 10
(B) 14
(C) 15
(D) 29
(E) 30


Sol:

All the numbers have even number of divisors except prime numbers and squares of prime numbers.

so the prime numbers whose square are less than 900 are : 2,3,5,7,11,13,17,19,23,29. count=10.
Option A is correct
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   10 Jul 2019, 06:24
A square of a prime no will be divisible by only 1, prime no and itself that is square of prime no.
Thus 2,3,5,7,11,13,17,19,23 and 29 are the numbers as any number above 30 will have a square greater than 900.
Thus option A

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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   10 Jul 2019, 07:50
Only prime numbers perfect squares have three positive divisors .

example,
2,3,5
perfect squares= 4,9,25 ...all of these have only 3 factors 1,prime number,and number itself.

The question is till what limit do we need to find the prime numbers so that the square of that number is less than 900

clearly 30 square =900 so all prime numbers less than will fall into into our set.

so 2,3,5,7,11,13,17,19,23,29 = 10

Hope its helps!!
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   02 Sep 2020, 23:52
Number Property Concept:

the only way an Integer can have 3 +Positive Factors if the Prime Factorization of the Number is as follows:

N = (p)^2nd ----- where p = prime number

Total No. of Factors = (2 + 1) = 3 Total + Divisors

Another words, the Question is asking us to find all the Prime Numbers SQUARED that have a Result less than < 900

(30)^2 = 900

The Prime Numbers Less Than 30 are:

2 - 3 - 5 - 7 - 11 - 13 - 17 - 19 - 23 - 29

10 Possible Numbers
Answer A
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   14 Nov 2020, 17:39
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Bunuel wrote:
How many integers from 1 to 900 inclusive have exactly 3 positive divisors?

(A) 10
(B) 14
(C) 15
(D) 29
(E) 30



Square of prime numbers will have exactly 3 factors, the numbers are:
2,3,5,7,11,13,17,19,23, and 29 = 10

Ans. A
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   15 Dec 2022, 13:44
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Bunuel wrote:
How many integers from 1 to 900 inclusive have exactly 3 positive divisors?

(A) 10
(B) 14
(C) 15
(D) 29
(E) 30


Only perfect squares that originate from a prime number contain three factors. Numbers such as 4 (with factors of 4, 2, and 1) or 9 (with factors of 9, 3, and 1).

So, let’s find the largest perfect square less than or equal to 900 that stems from a prime number. Since 30^2 = 900, 29 must be the largest perfect square in the set. Thus, the rest of the numbers in the set are:

23, 19, 17, 13, 11, 7, 5, 3, and 2.

So, there are ten numbers between 1 and 900 inclusive that have three factors.

Answer: A
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Re: How many integers from 1 to 900 inclusive have exactly 3 positive divi [#permalink] New post   20 Jan 2024, 10:08
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