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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Everest
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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.
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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stormier
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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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prachich1987
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The positive divisors of 4 are 1,2,4........three divisorsstormier 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 9 are 1,3,9....total three divisors.
hope it helps
10 is the correct answer.
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Everest
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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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thebigkats
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Hi:
how about numbers like 35 which have 1, 3 and 5 as divisors?
thanks
how about numbers like 35 which have 1, 3 and 5 as divisors?
thanks
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Brent@GMATPrepNow
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The divisor of 35 are: 1, 5, 7, 35thebigkats wrote:Hi:
how about numbers like 35 which have 1, 3 and 5 as divisors?
thanks
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The only integers that have exactly 3 positive divisors are perfect squares.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.
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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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
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