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How many odd positive integers divide the positive integer n completel

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How many odd positive integers divide the positive integer n completel  [#permalink]

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New post 26 Jul 2017, 03:15
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Question Stats:

56% (01:11) correct 44% (01:02) wrong based on 18 sessions

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How many odd positive integers divide the positive integer n completely?

(1) 16 is the highest power of 2 that divides n
(2) n has a total of 68 factors and 3 prime factors.

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How many odd positive integers divide the positive integer n completel  [#permalink]

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New post 26 Jul 2017, 05:40
niteshwaghray wrote:
How many odd positive integers divide the positive integer n completely?

(1) 16 is the highest power of 2 that divides n
(2) n has a total of 68 factors and 3 prime factors.



hi

1) 16 is the highest power of 2 that divides n
so \(n=2^{16}*a^x.....\)
insuff

(2) n has a total of 68 factors and 3 prime factors.
so 3 prime factors ..
all ODD, 68 factors is answer..
If one is EVEN that is 2, ans will be different..
insuff

combined
\(n=2^{16}*odd^x*odd^y\)...
now 68=17*2*2 or 17*4*1..
but only 17*2*2 is possible otherwise only 2 prime factors would be there


So number of factors = (16+1)(1+1)(1+1)..
so
number of ODD factors = 2*2=4
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Re: How many odd positive integers divide the positive integer n completel  [#permalink]

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New post 26 Jul 2017, 06:33
The sentence: "16 is the highest power of 2 that divides n" means "2^4 is the highest power of 2 that divides n". It does not mean "2^16 is the highest power of 2 that divides n", which is what the question writer intended. So the question is not worded properly.

Chetan above has figured out what the question writer meant, and his solution is perfect. Statement 1 tells us nothing about odd divisors, while Statement 2 doesn't tell us only that we have 68 divisors in total, but some of those might be even. Taken together, the number must be 2^16 * p^a * q^b, and using the familiar technique to count divisors, (16 + 1)(a + 1)(b + 1) = 68, so (a+1)(b+1) = 4, which means a and b are both 1. So we can find the exponents on the odd primes in the prime factorization of n, and that ensures we can count the odd divisors of n.
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Re: How many odd positive integers divide the positive integer n completel  [#permalink]

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New post 26 Jul 2017, 14:05
Ans C:
St. 1 or St. 2 is insufficient, when applied individually.

Using St 1 + 2:
N= (2^16) x (P1^1) x (P2^1)

Such that
* P1 and P2 are two distinct prime numbers other than 2, hence both are odd intergers, and
*Total factors = 17x2x2=68.

So, total odd divisors of N are 4.
They are P1, P2, P1xP2 and 1.

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This discussion does not meet community quality standards. It has been retired.


If you would like to discuss this question please re-post it in the respective forum. Thank you!

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Re: How many odd positive integers divide the positive integer n completel   [#permalink] 26 Jul 2017, 14:05
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