Question Pack 1 = If X is an integer greater than 1, is X eq

[richachampion]

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

[GMATGuruNY]

I posted an explanation here:
https://www.beatthegmat.com/interget-pow ... 14665.html

[Matt@VeritasPrep]

Try some small numbers to show that each statement alone is insufficient. (If x = 1, the answer is YES, if x = 2, the answer is NO.)

From there, we've got C or E.

If we know that x = m³ and and that x = n�, we can say that m³ = n�. That means that whatever x is, each of its prime factors can be broken up into THREES (one for each cube root) and into FOURS (one for each fourth root).

Since x's # of each prime factor is a multiple of 3 and of 4, it must be a multiple of the LCM of 3 and 4, or a multiple of 12. So x has 12 of each of its prime factors, and MUST be a twelfth power.

[Jay@ManhattanReview]

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

It is clear that each statement is insufficient.

Now combine the two.

S1: Say, X = K^(1/3) => X^3 = K; K is an integer;

S2: Say, X = M^(1/4) => X^4 = M; M is an integer;

=> X^3 * X^4 = K*M = N; N is an integer (Product of two integers, K and M is integer)

=> K^12 = N

=> X = N^(1/12)

Hope this helps!

-Jay

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[Mo2men]

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

It is clear that each statement is insufficient.

Now combine the two.

S1: Say, X = K^(1/3) => X^3 = K; K is an integer;

S2: Say, X = M^(1/4) => X^4 = M; M is an integer;

=> X^3 * X^4 = K*M = N; N is an integer (Product of two integers, K and M is integer)

=> K^12 = N

=> X = N^(1/12)

You have interpreted each statement as 3rd root and 4th root, while it is should be power.

x = K^3

x= M^4

[GMATGuruNY]

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, 2�, 2�, 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 = 2�, 2�, 2¹²...
If x = 2�, 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 = 2¹�, 2¹�, 2²¹, 2²�...
x = 2¹�, 2²�, 2²�...
The next value common to both lists is 2²� = 4¹², which is the 12th power of an integer.
Implication:
To satisfy both statements, x must be the 12th power of an integer.
SUFFICIENT.

The correct answer is C.

[Scott@TargetTestPrep]

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. We can eliminate answer choices A and D.

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. We can eliminate answer choice B.

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 fourth power, the fourth root of a^3 is an integer. The only way this could happen is if a is also the fourth power of an integer; in other words, a by itself is a fourth 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.

Answer: C

[Jay@ManhattanReview]

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

It is clear that each statement is insufficient.

Now combine the two.

S1: Say, X = K^(1/3) => X^3 = K; K is an integer;

S2: Say, X = M^(1/4) => X^4 = M; M is an integer;

=> X^3 * X^4 = K*M = N; N is an integer (Product of two integers, K and M is integer)

=> K^12 = N

=> X = N^(1/12)

You have interpreted each statement as 3rd root and 4th root, while it is should be power.

x = K^3

x= M^4

You rightly pointed out. Pl. read '^(1/3)' as '^3', '^(1/4)' as '^4', and '^(1/12)' as '^12', then the explanation is good to go. I liked Mitch's simplistic approach. It's better.