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3 positive Divisors

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3 positive Divisors [#permalink]

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New post 22 Mar 2011, 08:46
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How many integers from 1 to 900 inclusive have exactly 3 positive divisors?

1. 10
2. 14
3. 15
4. 29
5. 30

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Re: 3 positive Divisors [#permalink]

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New post 22 Mar 2011, 11:15
Now, if N = a^p x b^q x c^r ... where a, b, c... are prime numbers and p, q, r are positive integers, then number of factors of N is (p+1)(q+1)(r+1)...

In this case, Number of factors is 3 and two of the three factors of a number are 1 and the number itself. The third factor has to be a prime number.

P+1 = 3 or p = 2

Therefore, N = a^2

N < 900

or a^2 < 900

or a < 30

Since P is a prime number, p can assume 10 values (Take every prime number between 1 and 30 viz. 2, 3, 5, 7, 11, 13, 17, 19, 23 & 29)

Thus Ans = 10
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Re: 3 positive Divisors [#permalink]

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New post 22 Mar 2011, 11:20
All prime number have 2 divisors. 1 & the number it self.

So only Square of prime numbers (p) will have 3 divisors.
i.e. 1, p, p*p.

So the total number is count of those prime numbers whos square in the given range.

They are 2 3 5 7 11 13 17 19 23 29.

Total count 10 -- > choice A
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Re: 3 positive Divisors [#permalink]

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New post 27 Apr 2011, 07:29
Hi:
What about numbers like 35 - 1*3*5 - whcih also have 3 divisors but aren't squares of a prime no?

please help

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Re: 3 positive Divisors [#permalink]

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New post 27 Apr 2011, 07:59
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thebigkats wrote:
Hi:
What about numbers like 35 - 1*3*5 - whcih also have 3 divisors but aren't squares of a prime no?

please help


35=7^1*5^1.
Thus it will have (1+1)*(1+1)=4 Divisors

Check the section: "Finding the Number of Factors of an Integer" in the following link.
math-number-theory-88376.html

35 = 1*35
35 = 7*5

Total 4 divisors: 1,7,5,35.

All positive integers have even number of factors(or divisors) except perfect squares.

1: Perfect Square: 1^2: Factors=1 (Number of factors=1, Odd)
2: Not a perfect Square: Factors=1,2 (Number of factors=2, Even)
6: Not a perfect Square: Factors=1,2,3,6 (Number of factors=4, Even)
9: A perfect Square: 3^2; Factors=1,3,9 (Number of factors=3, Odd)
12: Not a perfect Square: Factors=1,2,3,4,6,12 (Number of factors=6, Even)
16: A perfect Square: 4^2; Factors=1,2,4,8,16 (Number of factors=5, Odd)

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

3=odd. We know that the number of factors is odd. Thus, the question is asking us to find the perfect squares as only perfect squares can have odd numbers of factors.

But, that's not all. Reason: We need the perfect squares with only 3 factors.
16: A perfect Square: 4^2; Factors=1,2,4,8,16 (Number of factors=5, Odd but not 3)
We see that 16 has 5 factors; we need only 3 factors. The reason is: the base(4) can be factored further resulting in more number of factors.

However, if we choose just the prime number as base and square them. The base can't be factorized further resulting in exactly 3 factors.

2^2=4; Factors: 1,2,4(Count=3) because 2(base) can't be factored.
3^2=9; Factors: 1,3,9(Count=3) because 3(base) can't be factored.
5^2=25; Factors: 1,5,25(Count=3) because 5(base) can't be factored.
13^2=169. Factors: 1,13,169(Count=3) because 13(base) can't be factored.

Thus, we need to find the number of primes between 1 and 900, such that its square shouldn't exceed 900.

1^2<(Prime Number)^2<900
OR
1^2<(Prime Number)^2<(30)^2
OR
1<Prime Number<30

Numbers that fit this range:
2,3,5,7,11,13,17,19,23,29(Count=10)

Ans: "A"
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Re: 3 positive Divisors [#permalink]

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New post 29 Apr 2011, 23:25
It has to be essentially a perfect square of prime number.

29 ^2 = 30^2 - (30+29) <900
Hence the max value of prime number is 29. Counting from 2 to 29 gives 10 prime numbers. Hence A.
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Re: 3 positive Divisors [#permalink]

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New post 08 May 2011, 03:43
For each factor of n (apart from 1 and itself), there must be a counter-factor. In other words, if x is a factor of n, so is n/x. So effectively for every new factor you add, you have an additional counter-factor coming in unless both the factor and its counter are equal (which is the case where n is a perfect square). Hope this helps!

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Re: 3 positive Divisors [#permalink]

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Re: 3 positive Divisors   [#permalink] 13 Aug 2017, 10:47
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