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# Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81

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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
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Kudos
Bunuel wrote:
If $$3^{(-\frac{k}{2})} > \frac{1}{81}$$, what is the value of k ?

(1) $$k$$ is the square of a prime number.

(2) $$\frac{k}{2}$$ is a prime number.

 This question was provided by GMAT Club for the Around the World in 80 Questions Win over $20,000 in prizes: Courses, Tests & more We know that 1/81 = 1/3^4 = 3^-4 Expressing both sides in terms of power of 3: -k/2 > -4 k/2 < 4 k < 8 From (1), If k is the square of a prime number, k = 2^2, 3^2, 5^2, .... = 4, 9, 25, ..... But from the given statement, k < 8, hence k = 4 is the only possible solution. (1) is sufficient From (2), k/2 is a prime number. So k = 4, 6, 10, .... so that k/2 = 2, 3, 5, .... (primes) Not sufficient Ans. (1) Manager Joined: 07 May 2023 Posts: 201 Own Kudos [?]: 243 [1] Given Kudos: 47 Location: India Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink] 1 Kudos Bunuel wrote: If $$3^{(-\frac{k}{2})} > \frac{1}{81}$$, what is the value of k ? (1) $$k$$ is the square of a prime number. (2) $$\frac{k}{2}$$ is a prime number.  This question was provided by GMAT Club for the Around the World in 80 Questions Win over$20,000 in prizes: Courses, Tests & more

$$3^{(-\frac{k}{2})} > \frac{1}{81}$$

$$3^{(-\frac{k}{2})} > 3^{-4}$$

$$-\frac{k}{2} < -4$$

$$- k > -8$$

$$k < 8$$

1) Only possible value of k = 4

The statement is sufficient.

2) k can be 4 or 6 as 4/2 = 6 and 6/2 = 3

The statement is not sufficient.

IMO A
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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
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Kudos
$$3^{(-\frac{k}{2})} > \frac{1}{81}$$

$$\frac{1}{3^\frac{k}{2}} > \frac{1}{3^4}$$

As we are dealing with fractions, for the left hand side to be a bigger we need the denominator to be smaller. Therefore with dropping the base of 3: $$\frac{k}{2}<4$$
$$k<8$$

(1) k is the square of a prime number.

Given the parameters of k, the only possible answer for k is 2. $$2^2 = 4$$ works. $$3^2 = 9$$ exceeds the inequality $$k<8$$

SUFFICIENT

(2) k/2 is a prime number.

If k is 6: $$\frac{6}{2}= 3$$
If k is 4: $$\frac{4}{2}= 2$$

One cannot come to a single value for k.

INSUFFICIENT

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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
1
Kudos
Given:
If 3^(−k/2)>1/81, what is the value of k ?
(1) k is the square of a prime number.
(2) k/2 is a prime number.

Let's evaluate:
=> 3^(-k/2) > 1/(3^4)
=> 3^(-k/2) > 3 (-4)

Therefore, we can write:
=> - k/2 = -4
=> -k > -8
=> k < 8

Statement 1: k is the square of a prime number.
2^2 = 4
3^2 = 9
So, here, we can definitely say that k is 4. (k<8)
Also, we should remember that square of any number is a positive number.

(Sufficient)

Statement 2:
k/2 is a prime number.

Plug in a few values:
k = 4 => 4/2 = 2 (prime)
k = 6 => 6/2 = 3 (prime)
So we don't have a single answer.
(Not sufficient.)

So that means, Statement 1 alone is sufficient.
I will go with option A.
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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
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Kudos
3^ (−k/2)> 1/81
3^ (−k/2)> 3^(-4)
(−k/2)> (-4)
k/2 < 4
k < 8

(1) k is the square of a prime number.
2^2 = 4

(2) k / 2 is a prime number.
6/2 = 3
4/2 = 2

So, 1 alone is sufficient, but 2 alone is not
Ans : A
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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
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Kudos
So here is my ans:

When we solve the equation we get k< 8

a) k is square of prime number

sqre of 2 = 4
sqre of 3 = 9

so we know that there can be only one option which is k=4

b) k/2 is prime number
so that means k is a even number
so k can be 4,6
when k = 4 =>4/2=2 prime number
when k = 6 =>6/2=3 prime number
so both the answers are possible.

Hence only a is sufficient
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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
1
Kudos
If 3^(-k/2) > 1/81. Then K=?

We can rewrite Question as:-

3^(-k/2) > 1/3^4
3^(-k/2 + 4) > 1
so for 3^x to be greater than 1. x should be greater than 0.

-k/2 + 4 > 0
k < 8
Question :- if K<8 what is k?

Statement 1: k is the square of a prime number.

i.e. K can take the values -> 2^2 , 3^2 , 5^2....... -> 4,9,25...
As K<8 only possible value is 4 so S1: SUFFICIENT

Statement 2: k/2 is a prime number.

i.e. K can take the values -> 2*2 , 3*2 , 5*2....... -> 4,6,10...
As K<8 k can take 4 and 6 so S2: INSUFFICIENT

IMO A
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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
1
Kudos
We can write 3^(-k/2) = 1/(3^(k/2))

Hence, 1/(3^(k/2)) > 1/81.

We can cross multiple the denominators for simplification and this wont change the sign between them, since "81" is positive and "3^(k/2)" is also positive (independent whether "k" is positive or negative)

Therefore:- 3^(k/2) < 81
So, (k/2) < 4, hence k < 8

(1) k is square of a prime number:-
Square of prime numbers are 4, 9, 25,.... and only "4" satisfies our condition of k < 8.

Hence statement (1) alone is sufficient.

(2) k/2 is a prime number:-
Using this condition, the possible values of "k" are 4, 6, 10, 14,....
Here "4" and "6" both satisfies the condition of k < 8. So using statement (2) we still cannot uniquely find the value of "k".

Hence statement (2) alone is NOT sufficient.
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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
1
Kudos
Bunuel wrote:
If $$3^{(-\frac{k}{2})} > \frac{1}{81}$$, what is the value of k ?

(1) $$k$$ is the square of a prime number.

(2) $$\frac{k}{2}$$ is a prime number.

 This question was provided by GMAT Club for the Around the World in 80 Questions Win over \$20,000 in prizes: Courses, Tests & more

Q:- $$3^{(-\frac{k}{2})} > 3^{(-4)}$$
-k/2 > -4
k/2 < 4
k < 8

s1 :- k = square of a prime number
prime number less than \sqrt{8} = 2
So, k = 2
Therefore, statement 1 is sufficient.

s2:- k/2 = prime number

prime numbers valid for above cases = 2 , 3
Therefore, s2 is not sufficient.

Hence, A is correct answer.
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Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
1
Kudos
Bunuel wrote:
If $$3^{(-\frac{k}{2})} > \frac{1}{81}$$, what is the value of k ?

$$3^{(-\frac{k}{2})} > \frac{1}{81}$$
=> $$3^{(-\frac{k}{2})}$$ > $$3^(-4)$$

=> $$-(\frac{k}{2})$$ > -4
=> $$\frac{k}{2 }$$< 4
=> k < 8

Quote:
(1) $$k$$ is the square of a prime number.

From Statement 1, we get
Squares of prime numbers are $$2^2$$, $$3^2$$, $$5^2$$ , and so on
But k < 8 , this implies that k = 4.

Statement 1 is sufficient

Quote:
(2) $$\frac{k}{2}$$ is a prime number.

From Statement 2, we get

$$\frac{k}{2}$$ = Prime number = n
=> k = 2 * n
Since k < 8
=> 2*n < 8
=> n < 4
Therefore, n could be 2 or 3.
Statement 2 is not sufficient.

IMO OA should be A.
Re: Around the World in 80 Questions (Day 2): If 3^(-k/2) > 1/81 [#permalink]
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