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# If n is a positive integer less than 50, what is the value

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If n is a positive integer less than 50, what is the value [#permalink]

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05 Nov 2006, 15:06
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If n is a positive integer less than 50, what is the value of n?

(1) n is a multiple of the square of the sum of two distinct factors of n.
(2) n is a perfect square with 9 factors.
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05 Nov 2006, 15:55
But it's a multiple of the square of the SUM of two distinct factors..
(1) is insufficient because, for example, (1+2)^2 = 9 so 18 and 36 work.
(2) hmm, 1 (no), 4 (no), 9 (no), 16 (no), 25 (no), 36 (yes), 49 (no) so sufficient. (1,2,3,4,6,9,12,18, 36 are factors of 36).
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07 Nov 2006, 06:04
joeydvivre wrote:
But it's a multiple of the square of the SUM of two distinct factors..
(1) is insufficient because, for example, (1+2)^2 = 9 so 18 and 36 work.
(2) hmm, 1 (no), 4 (no), 9 (no), 16 (no), 25 (no), 36 (yes), 49 (no) so sufficient. (1,2,3,4,6,9,12,18, 36 are factors of 36).

For the same reason B.

I am still a bit apprehensive about choosing B. Can a number be more than 50, perfect square, and has 9 factors? remote chance.
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the way is difficult, but perfect trust makes it easy.

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07 Nov 2006, 06:33
ak_idc wrote:
joeydvivre wrote:
But it's a multiple of the square of the SUM of two distinct factors..
(1) is insufficient because, for example, (1+2)^2 = 9 so 18 and 36 work.
(2) hmm, 1 (no), 4 (no), 9 (no), 16 (no), 25 (no), 36 (yes), 49 (no) so sufficient. (1,2,3,4,6,9,12,18, 36 are factors of 36).

For the same reason B.

I am still a bit apprehensive about choosing B. Can a number be more than 50, perfect square, and has 9 factors? remote chance.

Remote chance? There are an infinite number of them.
In fact, if n = p1*p2 where p1 and p2 are prime numbers, then n^2 has 9 factors. The factors are {1, p1, p2, p1^2, p1*p2, p2^2, p1*p2^2, p2*p1^2, n^2}. Examples of such numbers are
100 = {1, 2, 4, 5, 10, 20, 25, 50, 100}
225 = {1, 3, 5, 9, 15, 25, 45, 75, 225}
....
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08 Nov 2006, 05:50
joeydvivre wrote:
ak_idc wrote:
joeydvivre wrote:
But it's a multiple of the square of the SUM of two distinct factors..
(1) is insufficient because, for example, (1+2)^2 = 9 so 18 and 36 work.
(2) hmm, 1 (no), 4 (no), 9 (no), 16 (no), 25 (no), 36 (yes), 49 (no) so sufficient. (1,2,3,4,6,9,12,18, 36 are factors of 36).

For the same reason B.

I am still a bit apprehensive about choosing B. Can a number be more than 50, perfect square, and has 9 factors? remote chance.

Remote chance? There are an infinite number of them.
In fact, if n = p1*p2 where p1 and p2 are prime numbers, then n^2 has 9 factors. The factors are {1, p1, p2, p1^2, p1*p2, p2^2, p1*p2^2, p2*p1^2, n^2}. Examples of such numbers are
100 = {1, 2, 4, 5, 10, 20, 25, 50, 100}
225 = {1, 3, 5, 9, 15, 25, 45, 75, 225}
....

_________________

The path is long, but self-surrender makes it short;
the way is difficult, but perfect trust makes it easy.

08 Nov 2006, 05:50
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