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If the sum of the positive factors of a Positive integer N is P and th

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GMATinsight
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If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   22 Feb 2021, 06:20
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If the sum of the positive factors of a Positive integer N is P and the product of all positive factors of N is Q. Then how many values of N exist from 1 to 50 (both inclusive) such that P*Q = ODD?

A) 5
B) 6
C) 7
D) 8
E) 9



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If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   22 Feb 2021, 08:57
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GMATinsight wrote:
If the sum of the positive factors of a Positive integer N is P and the product of all positive factors of N is Q. Then how many values of N exist from 1 to 50 (both inclusive) such that P*Q = ODD?

A) 5
B) 6
C) 7
D) 8
E) 9


P*Q = Odd. => So both P and Q should be odd.

Q is the product of factors and Q is odd. It is possible only when all factors are odd : Odd*odd*odd....=odd
If the factors are odd, the number itself has to be ODD.

Now sum of the positive factors, all of which are odd, is odd.
That is odd+odd+odd+...=odd.
This will be the case only when the number of factors are odd.

If the number of factors are odd, the number N will be a square.

So N is odd and a perfect square.

Squares of odd integers from 1 to 50 : Till 7^2 or 49
So 1, 3, 5 and 7.

4 is the answer.
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Re: If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   23 Feb 2021, 00:20
question here is testing + factors of numbers from 1 to 50 inclusive
such that the result of factors P*Q is odd ; which is possible only when the value of both P & Q is odd
we can eliminate option which are even i.e. B, D
also we can see that the square of the numbers from 1 to 7 falls into range 1 to 50 i.e. the square of integer always has odd set of integers.
so total values of N would be 7
option C


GMATinsight wrote:
If the sum of the positive factors of a Positive integer N is P and the product of all positive factors of N is Q. Then how many values of N exist from 1 to 50 (both inclusive) such that P*Q = ODD?

A) 5
B) 6
C) 7
D) 8
E) 9



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Re: If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   08 Mar 2021, 22:05
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chetan2u wrote:
GMATinsight wrote:
If the sum of the positive factors of a Positive integer N is P and the product of all positive factors of N is Q. Then how many values of N exist from 1 to 50 (both inclusive) such that P*Q = ODD?

A) 5
B) 6
C) 7
D) 8
E) 9


P*Q = Odd. => So both P and Q should be odd.

Q is the product of factors and Q is odd. It is possible only when all factors are odd : Odd*odd*odd....=odd

Now sum of the positive factors, all of which are odd, is odd.
That is odd+odd+odd+...=odd.
This will be the case only when the number of factors are odd.

If the number of factors are odd, the number N will be a square.
Squares from 1 to 50 : Till 7^2 or 49
So 7 such numbers.

C


But if N = 4, we have sum (P) = odd (1,2,4) and product (Q) = even
In that case P*Q = even

Same with N = 36, sum = (1,3,4,12,36) = even; Product = even (snic we have atleast 1 even), therefore, P*Q = even

Shouldn't the solution then exclude even squares?

Or am i missing something?
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Re: If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   09 Mar 2021, 00:14
Yes seems not to work for perfect squares of even numbers because while sums of factors odd, products must be even. So all I got was 1, 9, 25, and 49. Please correct me if I'm wrong but seeing ravigupta2912's point here.

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Re: If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   09 Mar 2021, 01:32
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ravigupta2912 wrote:
chetan2u wrote:
GMATinsight wrote:
If the sum of the positive factors of a Positive integer N is P and the product of all positive factors of N is Q. Then how many values of N exist from 1 to 50 (both inclusive) such that P*Q = ODD?

A) 5
B) 6
C) 7
D) 8
E) 9


P*Q = Odd. => So both P and Q should be odd.

Q is the product of factors and Q is odd. It is possible only when all factors are odd : Odd*odd*odd....=odd

Now sum of the positive factors, all of which are odd, is odd.
That is odd+odd+odd+...=odd.
This will be the case only when the number of factors are odd.

If the number of factors are odd, the number N will be a square.
Squares from 1 to 50 : Till 7^2 or 49
So 7 such numbers.

C


But if N = 4, we have sum (P) = odd (1,2,4) and product (Q) = even
In that case P*Q = even

Same with N = 36, sum = (1,3,4,12,36) = even; Product = even (snic we have atleast 1 even), therefore, P*Q = even

Shouldn't the solution then exclude even squares?

Or am i missing something?



Yes, you are correct.
Although I have written Q as odd*odd*odd*..., I didn’t apply that in the final step.
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Re: If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   09 Mar 2021, 07:00
The correct answer should then be 3, since only 3,5, and 7 fit in the conditions. And that’s not there in the options. Please correct me if wrong.

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Re: If the sum of the positive factors of a Positive integer N is P and th [#permalink] New post   13 Mar 2021, 16:39
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GMATinsight wrote:
If the sum of the positive factors of a Positive integer N is P and the product of all positive factors of N is Q. Then how many values of N exist from 1 to 50 (both inclusive) such that P*Q = ODD?

A) 5
B) 6
C) 7
D) 8
E) 9


Solution:

Since P*Q is odd, P and Q must be both odd. Since Q is the product of the factors of N and it is odd, then all the factors of N must be odd. In other words, N must be odd. Furthermore, since P is the sum of the factors of N and it is odd, then there must be an odd number of factors of N. With this in mind, let k be the number of factors of N.

If k = 1, then N must be 1 so that P = 1 and Q = 1 (notice that P*Q = 1).

If k = 3, then N could be 3^2, 5^2, or 7^2. (For example, if N = 5^2, P = 1 + 5 + 25 = 31 and Q = 1 x 5 x 25 = 125.)

If k = 5, then the minimum value of N is 3^4 = 81. However, this is greater than 50 already.

Answer: 4
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Re: If the sum of the positive factors of a Positive integer N is P and th [#permalink]
13 Mar 2021, 16:39
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