Three numbers x, y, and z are positive integers. What is the exact value of x? Statement I… — State PSC Exam

Three numbers $x$ , $y$ , and $z$ are positive integers. What is the exact value of $x$ ?

Statement I: The arithmetic mean of $x$ , $y$ , and $z$ is equal to $x + 2$ , where $y \le z$ .
Statement II: The least common multiple (LCM) of $y$ and $z$ is $12$ , and their greatest common divisor (GCD) is $2$ .

Which of the following options correctly describes the sufficiency of the statements?

A
Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.
B
Statement II alone is sufficient to answer the question, but Statement I alone is not sufficient.
C
Either Statement I alone or Statement II alone is sufficient to answer the question.
Neither Statement I nor Statement II is sufficient to answer the question, even when taken together.Answer
E
Both Statement I and Statement II together are sufficient to answer the question, but neither statement alone is sufficient.

Answer

Neither Statement I nor Statement II is sufficient to answer the question, even when taken together.

The correct answer states that neither statement is sufficient, even when combined. Statement I reduces to

y + z = 2x + 6

, which leaves

x

dependent on the sum

y + z

. Statement II yields two possible pairs for

(y, z)

with

\text{GCD}=2

and

\text{LCM}=12

, namely

(2, 12)

with sum

14

, and

(4, 6)

with sum

10

. Substituting

y + z = 14

gives

x = 4

, while substituting

y + z = 10

gives

x = 2

. Because both

x = 4

and

x = 2

are valid positive integer solutions, a single unique value for

x

cannot be determined.

Step-by-Step Solution

Analyze Statement I individually.

The arithmetic mean formula gives

\frac{x + y + z}{3} = x + 2 \implies x + y + z = 3x + 6 \implies y + z = 2x + 6

Since

y

and

z

are unknown positive integers,

x

can take multiple values (e.g., if

y+z = 8

x=1

; if

y+z = 10

x=2

). Thus, Statement I alone is insufficient.

Analyze Statement II individually.

Find pairs

(y, z)

such that

y \le z

\text{GCD}(y, z) = 2

, and

\text{LCM}(y, z) = 12

Since

y \times z = \text{GCD} \times \text{LCM} = 24

, the valid pairs of positive integers with

\text{GCD}=2

and

\text{LCM}=12

are

(2, 12)

and

(4, 6)

. Statement II makes no reference to

x

, so it is insufficient.

Evaluate Statements I and II together.

Test both valid pairs from Statement II in the relation

2x = y + z - 6

from Statement I.

Case 1: For

(y, z) = (2, 12)

y + z = 14 \implies 2x = 14 - 6 = 8 \implies x = 4

.
Case 2: For

(y, z) = (4, 6)

y + z = 10 \implies 2x = 10 - 6 = 4 \implies x = 2

.
Since two distinct values of

x

(

x = 4

and

x = 2

) satisfy all conditions, both statements together are NOT sufficient to determine a unique value for

x

Key Concept

Data Sufficiency with Multiple Discrete Solutions in Number Theory

Question

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