Value of X

[vinay1983]

What is the value of X, given X is a positive number?

1. X^3 is a 2 digit positive odd integer
2. X^4 is a 2 digit odd integer.

A B C D E

[Brent@GMATPrepNow]

What is the value of X, given X is a positive number?

1. X^3 is a 2 digit positive odd integer
2. X^4 is a 2 digit odd integer.

IMPORTANT: (nth root of k)^n = k
Examples: (square root of 9)^2 = 9
(square root of 22)^2 = 22
(cube root of 64)^3 = 64
(fourth root of 19)^2 = 19

Okay, onto the question. . . .

Target question: What is the value of X?

Given: X > 0

Statement 1: X^3 is a 2 digit positive odd integer
There are many values of X that meet this condition. Here are two:
Case a: X = 3, since 3^3 = 27, a 2-digit number.
Case b: X = cuberoot of 11, since (cuberoot of 11)^3 = 11, a 2-digit number.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: X^4 is a 2 digit odd integer.
There are many values of X that meet this condition. Here are two:
Case a: X = 3, since 3^4 = 81, a 2-digit number.
Case b: X = fourth root of 11, since ( fourth root of 11)^3 = 11, a 2-digit number.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
From both statements combined, we can conclude that X is an integer. Here's why?
We know this because X^3 and X^4 are both integers.
Notice that X^4 = (X^3)(X)
Let's assume for a moment that X is a non-integer. If X^3 is an integer, and X is a non-integer, then their product (X^4) cannot be an integer. So, X must be an integer.
If X is an integer, X must equal 3, since this is the only integer that yields 2-digit numbers when cubed and raised to the fourth power.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

[GMATGuruNY]

What is the value of X, given X is a positive number?

1. X^3 is a 2 digit positive odd integer
2. X^4 is a 2 digit odd integer.

Notice when the question stem employs the word NUMBER -- an indication that we should consider NON-INTEGER values.

Statement 1: x³ is a 2-digit positive odd integer
It's possible that x=3, since 3³ = 27.
It's possible that x=11^(1/3), since [11^(1/3)]³ = 11.
Since x can be different values, INSUFFICIENT.

Statement 2: x� is a 2-digit odd integer.
It's possible that x=3, since 3� = 81.
It's possible that x=11^(1/4), since [11^(1/4)]� = 11.
Since x can be different values, INSUFFICIENT.

Statements combined:
The only rational value that satisfies both statements is x=3.
Any irrational value that yields an integer value when cubed -- such as 11^(1/3) -- will NOT yield an integer value when raised to the 4th power:
[11^(1/3)]� = 11^(4/3).
Thus, there is only ONE value that satisfies both statements:
x=3.
SUFFICIENT.

The correct answer is C.

[Brent@GMATPrepNow]

Hi Mitch,

I can't say I agree with your logic for the two statements combined (unless I'm missing something totally basic, which is quite possible ).
If it were the case that only two x values satisfied statement 1, and only two x values satisfied statement 2, then I'd agree that x must equal 3, since the two statements share this solution.
However, there are many values of x that satisfy each statement.

To illustrate what I mean, we can examine would happen if statement 1 stated that x^3 < 1000, and statement 2 stated that x^4 < 1000.
For statement 1, we could say that x = 3 or 4 (plus others), and for statement 2, we could say that x = 2 or 3 (plus others)
BUT, for the combined statements, we couldn't conclude that x must equal 3.

Cheers,
Brent

[GMATGuruNY]

Hi Mitch,

I can't say I agree with your logic for the two statements combined (unless I'm missing something totally basic, which is quite possible ).

Hi, Brent --

My reasoning was as follows:
While more than one value satisfies the constraint in statement 1 (x=3 and x=11^(1/3) are two examples), and more than one value satisfies the constraint in statement 2 (x=3 and x=11^(1/4) are two examples), only ONE value satisfies both constraints: x=3.
To avoid confusion, I've clarified the reasoning in my post above.