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

Question

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

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

RajatVerma1392 · Accepted Answer

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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firas92 · Answer

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)

Archit3110 · Answer

square of prime no will have exactly 3 +ve factors ; 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
IMO A: 10

abhishekdadarwal2009 · Answer

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

Vini07 · Answer

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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prabsahi · Answer

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!!

Fdambro294 · Answer

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

MHIKER · Answer

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

ScottTargetTestPrep · Answer

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

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

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