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Manager  S
Joined: 23 Jan 2016
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Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

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can the answer not be as simple as - to be the 12th power of an integer, x would need to be the 3rd power and 4th power as well. Since individually the statements do not tell us this information, if we combine we get that x= m^12. Experts could you please let me know if this approach is wrong.

Thanks.
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Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

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Prompt Analysis
x is an integer.

Superset
The answer to this question will be either yes or no.

Translation
In order to know if x = p^12, we need:
1# exact value of p or x
2# any property or equation to infer the statement.

Statement analysis

St 1: x =a^3. Insufficient as a may or may not be the form of b^4. Hence option a, d eliminated.
St 2: x = c^4. Insufficient as a may or may not be the form of d^4. Hence b is eliminated.

St 1 & St 2: LCM of 4 and 3 is 12. Hence x is an integer to the power 12. Option C.
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Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

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1
Hi, I need some help.

I follow the above posts on (i) and (ii) as not sufficient. However, I am struggling with a case where, if true, leads to the answer E, not C:

If $$x=m^3=n^4$$, where $$m=2^8$$ and $$n=2^6$$, then

$$x=(2^8)^3=(2^6)^4=2^{24}$$ which does NOT equal $$2^{12}$$; NOT SUFFICIENT

But where where $$m=2^4$$ and $$n=2^3$$, then

$$x=(2^4)^3=(2^3)^4=2^{12}$$, SUFFICIENT

How is my thinking flawed here?
Math Expert V
Joined: 02 Sep 2009
Posts: 58410
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

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2
DanielAustin wrote:
Hi, I need some help.

I follow the above posts on (i) and (ii) as not sufficient. However, I am struggling with a case where, if true, leads to the answer E, not C:

If $$x=m^3=n^4$$, where $$m=2^8$$ and $$n=2^6$$, then

$$x=(2^8)^3=(2^6)^4=2^{24}$$ which does NOT equal $$2^{12}$$; NOT SUFFICIENT

But where where $$m=2^4$$ and $$n=2^3$$, then

$$x=(2^4)^3=(2^3)^4=2^{12}$$, SUFFICIENT

How is my thinking flawed here?

If $$x=2^{24}$$ it's still is the 12th power of an integer: $$x=2^{24}=4^{12}$$
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Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

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bhupi wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer

(2) x is equal to the 4th Power of an integer.

We are given that x is an integer greater than 1 and must determine whether x is equal to the 12th power of an integer.

Statement One Alone:

x is equal to the 3rd power of an integer.

Using the information in statement one, we cannot determine whether x is equal to the 12th power of an integer. For example, if x = 8 = 2^3, then it’s not equal to the 12th power of an integer. However, if x = (2^4)^3 = 2^12, then it is equal to the 12th power of an integer. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x is equal to the 4th power of an integer

Using the information in statement two, we cannot determine whether x is equal to the 12th power of an integer. For example, if x = 16 = 2^4, then it’s not equal to the 12th power of an integer. However, if x = (2^3)^4 = 2^12, then it is equal to the 12th power of an integer. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the information from statements one and two, we know that x is equal to the 3rd power of an integer and that x is also equal to the 4th power of some other integer. Let’s represent x as a^3 where a is an integer > 1. Since a^3 is also a 4th power, the 4th root of a^3 is an integer. The only way this could happen is if a is also the 4th power of an integer; in other words, a by itself is a 4th power, say a = b^4 where b is an integer > 1.

Thus, x = a^3 = (b^4)^3 = b^12. Therefore, x is equal to the 12th power of an integer.

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Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

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I believe that for this question one has to realize that the number is both a power of 3 and 4 and being 12 the LCM it is indeed a power of 12th..
Bunuel wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer --> $$x=m^3$$ for some positive integer $$m$$. If $$m$$ itself is 4th power of some integer (for example if $$m=2^4$$), then the answer will be YES (since in this case $$x=(2^4)^3=2^{12}$$), but if it's not (for example if $$m=2$$), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that $$x^4=m^{12}$$.

(2) x is equal to the 4th Power of an integer --> $$x=n^4$$ for some positive integer $$n$$. If $$n$$ itself is 3rd power of some integer (for example if $$n=2^3$$), then the answer will be YES (since in this case $$x=(2^3)^4=2^{12}$$), but if it's not (for example if $$n=2$$), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that $$x^3=n^{12}$$.

(1)+(2) Divide (i) by (ii): $$x=(\frac{m}{n})^{12}=integer$$. Now, $$\frac{m}{n}$$ can be neither an irrational number (since it's the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore $$\frac{m}{n}$$ must be an integer, hence $$x=(\frac{m}{n})^{12}=integer^{12}$$. Sufficient.

Hope it's clear.

P.S. Please read and follow: http://gmatclub.com/forum/rules-for-pos ... 33935.html
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Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

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bhupi wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer

(2) x is equal to the 4th Power of an integer.

Test an EASY CASE.
Test POWERS OF 2.

Statement 1:
x = 2³, 2⁶, 2⁹, 2¹²...
If x = 2³, then x is NOT equal to the 12th power of an integer.
If x = 2¹², then x IS equal to the 12th power of an integer.
INSUFFICIENT.

Statement 2:
x = 2⁴, 2⁸, 2¹²...
If x = 2⁴, then x is NOT equal to the 12th power of an integer.
If x = 2¹², then x IS equal to the 12th power of an integer.
INSUFFICIENT.

Statements combined:
The smallest value common to both the red list and the blue list is 2¹², which is the 12th power of an integer.
If we extend the two lists, we get:
x = 2¹⁵, 2¹⁸, 2²¹, 2²⁴...
x = 2¹⁶, 2²⁰, 2²⁴...
The next value common to both lists is 2²⁴ = 4¹², which is the 12th power of an integer.
Implication:
To satisfy both statements, x must be the 12th power of an integer.
SUFFICIENT.

Alternate approach:

Statement 1: x = a³, where a is an integer[/b]
If a=2, then x = 2³, which is not the 12th power of an integer.
If a=2⁴, then x = (2⁴)³ = 2¹², which is the 12th power of an integer.
INSUFFICIENT.

Statement 2: x = b⁴, where b is an integer
If b=2, then x = 2⁴, which is not the 12th power of an integer.
If b=2³, then x = (2³)⁴ = 2¹², which is the 12th power of an integer.
INSUFFICIENT.

Statements 1 and 2 combined:

Since x = a³ and x = b⁴, we get:
a³ = b⁴
a³ = (b³)b
b = (a/b)³.
Since b is an integer, (a/b)³ is an integer.
Since a/b = integer/integer -- the definition of a rational number -- it is not possible that a/b is equal to an irrational value such as ³√2.
Thus, in order for (a/b)³ to be an integer, a/b must be an integer, implying that b is the CUBE OF AN INTEGER.
Thus, x = b⁴ = (integer³)⁴ = integer¹².
SUFFICIENT.

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Available for tutoring in NYC and long-distance. Re: If x is an integer greater than 1, is x equal to the 12th   [#permalink] 16 Oct 2018, 13:27

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