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by stormier » Fri Jan 28, 2011 6:26 pm
How many integers from 1 to 900 inclusive have exactly 3 positive divisors?


This is a BTG practice problem. The answer they have is 10, which I think is incorrect.
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by Everest » Fri Jan 28, 2011 8:37 pm
An integer has exactly 3 divisors if and only if it is the square of a prime number.
Below are the prime numbers which when squared result less than 900.

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

There are 10 integers which have exactly 3 positive divisors.
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by stormier » Sat Jan 29, 2011 5:45 am
Everest wrote:An integer has exactly 3 divisors if and only if it is the square of a prime number.
Below are the prime numbers which when squared result less than 900.

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

There are 10 integers which have exactly 3 positive divisors.

I think this is incorrect.

2^2 =4

the positive divisors of 4 are 1 and 2.

They are not 1, 2, 2

If I were to follow this logic, I could say that 1,2,2,2,2 are positive divisors of 4 and so there are 5 positive divisors!!!

If someone asks you, what are the divisors (factors) of 9, you say 1 and 3. you never say 1,3 and 3.

1,3 and 3 are not three, but two positive divisors of 9.

If these numbers are not unique then the question does not make sense. And, no number can have exactly 3 unique positive divisors.
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by prachich1987 » Sat Jan 29, 2011 10:21 am
stormier wrote:
Everest wrote:An integer has exactly 3 divisors if and only if it is the square of a prime number.
Below are the prime numbers which when squared result less than 900.

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

There are 10 integers which have exactly 3 positive divisors.

I think this is incorrect.

2^2 =4

the positive divisors of 4 are 1 and 2.

They are not 1, 2, 2

If I were to follow this logic, I could say that 1,2,2,2,2 are positive divisors of 4 and so there are 5 positive divisors!!!

If someone asks you, what are the divisors (factors) of 9, you say 1 and 3. you never say 1,3 and 3.

1,3 and 3 are not three, but two positive divisors of 9.

If these numbers are not unique then the question does not make sense. And, no number can have exactly 3 unique positive divisors.
The positive divisors of 4 are 1,2,4........three divisors
& the positive divisors of 9 are 1,3,9....total three divisors.

hope it helps

10 is the correct answer.
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by Everest » Sat Jan 29, 2011 11:39 am
stormier wrote:
Everest wrote:An integer has exactly 3 divisors if and only if it is the square of a prime number.
Below are the prime numbers which when squared result less than 900.

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

There are 10 integers which have exactly 3 positive divisors.

I think this is incorrect.

2^2 =4

the positive divisors of 4 are 1 and 2.

They are not 1, 2, 2

If I were to follow this logic, I could say that 1,2,2,2,2 are positive divisors of 4 and so there are 5 positive divisors!!!

If someone asks you, what are the divisors (factors) of 9, you say 1 and 3. you never say 1,3 and 3.

1,3 and 3 are not three, but two positive divisors of 9.

If these numbers are not unique then the question does not make sense. And, no number can have exactly 3 unique positive divisors.

If you understand the explantion clearly.

Here I am talking about the square of prime numbers have exactly 3 divisors (Prime numbers are 2, 3 , 5, 7, 11, 13, 17, 19, 23, 29)

i.e. 4, 9, 25, 49, 121, 169, 289, 361, 529, 841 (Since 31 and above result > 900 we can ignore them)

count of these numbers are 10.
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by stormier » Sun Jan 30, 2011 6:04 am
OK - I had my "duh" moment there ! Thank you.
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by thebigkats » Wed Apr 27, 2011 6:38 am
Hi:
how about numbers like 35 which have 1, 3 and 5 as divisors?
thanks
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by Brent@GMATPrepNow » Wed Apr 27, 2011 6:43 am
thebigkats wrote:Hi:
how about numbers like 35 which have 1, 3 and 5 as divisors?
thanks
The divisor of 35 are: 1, 5, 7, 35

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by GMATGuruNY » Wed Apr 27, 2011 6:51 am
stormier wrote:How many integers from 1 to 900 inclusive have exactly 3 positive divisors?


This is a BTG practice problem. The answer they have is 10, which I think is incorrect.
The only integers that have exactly 3 positive divisors are perfect squares.
If perfect square X is the square of prime number p, then X will have exactly 3 positive divisors: 1, p, and X.

In the problem above, since perfect square X < 900, we know that p^2 < 900.
Thus, p < 30.
Prime numbers less than 30: 2,3,5,7,11,13,17,19,23,29 = 10.
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