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# How many integers from 1 to 2003 inclusive have an odd number of disti

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How many integers from 1 to 2003 inclusive have an odd number of disti  [#permalink]

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26 Nov 2016, 02:54
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Difficulty:

55% (hard)

Question Stats:

61% (01:39) correct 39% (01:45) wrong based on 112 sessions

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How many integers from 1 to 2003 inclusive have an odd number of distinct positive divisors?

(A) 100
(B) 75
(C) 50
(D) 46
(E) 44

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How many integers from 1 to 2003 inclusive have an odd number of disti  [#permalink]

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26 Nov 2016, 03:19
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Bunuel wrote:
How many integers from 1 to 2003 inclusive have an odd number of distinct positive divisors?

(A) 100
(B) 75
(C) 50
(D) 46
(E) 44

If $$N=p_1^{q_1} \times p_2^{q_2} \times ... \times p_n^{q_n}$$, where $$p_1, p_2, ..., p_n$$ are distinct prime divisors, then the number of factors of $$N$$ is $$(1+q_1)(1+q_2)...(1+q_n)$$

If $$N$$ has an odd number of distinct divisors, then $$(1+q_1)(1+q_2)...(1+q_n)$$ is odd, so $$q_1, q_2, ..., q_n$$ are all even.

So, we simply count the number of integers from 1 to 2003 that these numbers are squared.
For example, $$1^2$$ has only 1 positive divisors.
$$2^2$$ has 3 positive divisors 1, 2, and 4.
$$3^2$$ has 3 positive divisors 1, 3, and 9.
$$4^2=2^4=16$$ has 5 positive divisors 1, 2, 4, 8, 16.
$$6^2 =2^2 \times 3^2=36$$ has 9 positive divisors 1, 2, 3, 4, 6, 9, 12, 18, 36.

Note that $$44^2=1936$$ and $$45^2=2025$$, so these numbers are $$1^2, 2^2,..., 44^2$$. There are 44 numbers like them.

In this solution, we count only squared number. What about other number is formed by even exponent like $$k^4, k^6, k^8,...$$?
Note that $$k^4=(k^2)^2$$, and we've already counted $$k^2$$. For example, counting $$2^4$$ means counting $$(2^2)^2=4^2$$.
Also, $$k^6=(k^3)^2$$. For example, counting $$3^6$$ means counting $$(3^3)^2=27^2$$.
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How many integers from 1 to 2003 inclusive have an odd number of disti  [#permalink]

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Updated on: 27 Nov 2016, 00:06
3
a perfect square always have odd number of distinct factors.

So the question is basically asking us to find how many numbers have got squares less than 2003.

Rest of the numbers exceed 2003.
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Originally posted by SI777 on 26 Nov 2016, 09:00.
Last edited by SI777 on 27 Nov 2016, 00:06, edited 1 time in total.
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How many integers from 1 to 2003 inclusive have an odd number of disti  [#permalink]

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26 Nov 2016, 14:16
This seems like a level 700 question

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Re: How many integers from 1 to 2003 inclusive have an odd number of disti  [#permalink]

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08 Apr 2018, 19:51
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Hello from the GMAT Club BumpBot!

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Re: How many integers from 1 to 2003 inclusive have an odd number of disti   [#permalink] 08 Apr 2018, 19:51
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