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# If Q the square of an odd positive integer and if 8Q^8 has four prime

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Math Expert
Joined: 02 Sep 2009
Posts: 60727
If Q the square of an odd positive integer and if 8Q^8 has four prime  [#permalink]

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25 Nov 2019, 01:20
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Difficulty:

65% (hard)

Question Stats:

49% (02:00) correct 51% (01:59) wrong based on 74 sessions

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If Q the square of an odd positive integer and if 8Q^8 has four prime factors, then how many prime factors does √Q have?

A. 1
B. 2
C. 3
D. 4
E. Cannot be determined

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VP
Joined: 19 Oct 2018
Posts: 1297
Location: India
Re: If Q the square of an odd positive integer and if 8Q^8 has four prime  [#permalink]

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25 Nov 2019, 03:41
As $$Q^2$$ is an odd number, $$Q^8$$ must be odd too.

$$2^3*Q^8$$ has 4 prime factors; hence, $$Q^8$$ must have 3 prime factors, and so does $$\sqrt{Q}$$.

Bunuel wrote:
If Q the square of an odd positive integer and if 8Q^8 has four prime factors, then how many prime factors does √Q have?

A. 1
B. 2
C. 3
D. 4
E. Cannot be determined

Are You Up For the Challenge: 700 Level Questions
Manager
Joined: 10 Dec 2017
Posts: 177
Location: India
If Q the square of an odd positive integer and if 8Q^8 has four prime  [#permalink]

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26 Nov 2019, 06:29
Bunuel wrote:
If Q the square of an odd positive integer and if 8Q^8 has four prime factors, then how many prime factors does √Q have?

A. 1
B. 2
C. 3
D. 4
E. Cannot be determined

Are You Up For the Challenge: 700 Level Questions

Q=(2n+1)^2
Q^8=(2n+1)^16
$$8*Q^8= 2^3*(2n+1)^(16)$$
This means 2n+1 consists of 3 prime factors as 2 is one prime factor and total prime factors in 8Q^8 is 4
So $$\sqrt{Q}$$ has 3 prime factors.
Intern
Joined: 17 Dec 2017
Posts: 9
Re: If Q the square of an odd positive integer and if 8Q^8 has four prime  [#permalink]

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13 Dec 2019, 10:43
Q8 must have 3 prime factors, and so does root over Q
can u plz explain this one
VP
Joined: 19 Oct 2018
Posts: 1297
Location: India
If Q the square of an odd positive integer and if 8Q^8 has four prime  [#permalink]

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13 Dec 2019, 16:29
1
harivars

$$8*Q^8= 2^{3+d}*a^x*b^y*c^z$$, where a, b and c are distinct odd prime numbers

As Q is square of odd integer, Q can't have 2 as its prime factor. d must be equal to 0.

so, $$Q^8 = a^x*b^y*c^z$$

$$[Q^{1/2}]^{16}= a^x*b^y*c^z$$

$$[Q^{1/2}]= a^{x/16}*b^{y/16}*c^{z/16}$$

$$Q^{1/2}$$ has 3 prime factors a, b and c, which are same as that of Q^8.

harivars wrote:
Q8 must have 3 prime factors, and so does root over Q
can u plz explain this one
If Q the square of an odd positive integer and if 8Q^8 has four prime   [#permalink] 13 Dec 2019, 16:29
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