What is the exact value of the two-digit positive integer N? Statement (I): The sum of the… — State PSC Exam

What is the exact value of the two-digit positive integer $N$ ?

Statement (I): The sum of the tens digit and the units digit of $N$ is equal to $12$ .
Statement (II): Reversing the digits of $N$ yields a new two-digit integer that is $36$ greater than $N$ .

Which of the following statements correctly describes the sufficiency of the given data to answer the question?

A
Statement (I) alone is sufficient, but Statement (II) alone is not sufficient.
B
Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.
C
Either Statement (I) alone or Statement (II) alone is sufficient.
D
Neither Statement (I) nor Statement (II) is sufficient.
Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.Answer

Answer

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.

Evaluating each statement independently reveals that neither statement alone narrows

N

down to a single value. Combining both equations (

x + y = 12

and

y - x = 4

) yields a single unique pair

x = 4

and

y = 8

, giving

N = 48

. Therefore, both statements together are required and sufficient.

Step-by-Step Solution

Represent the two-digit integer algebraically.

Let

N = 10x + y

, where

x

is the tens digit (

1 \leq x \leq 9

) and

y

is the units digit (

0 \leq y \leq 9

Setting up standard place-value variables allows formal analysis of both statements.

Evaluate Statement (I) alone.

From Statement (I),

x + y = 12

. Possible pairs

(x,y)

are

(3,9), (4,8), (5,7), (6,6), (7,5), (8,4), (9,3)

. Thus,

N

could be

39, 48, 57, 66, 75, 84,

93

Since multiple valid values of

N

exist, Statement (I) alone is NOT sufficient.

Evaluate Statement (II) alone.

The reversed number is

10y + x

. Given

(10y + x) - (10x + y) = 36 \implies 9(y - x) = 36 \implies y - x = 4

. Possible pairs

(x,y)

are

(1,5), (2,6), (3,7), (4,8), (5,9)

, giving

N = 15, 26, 37, 48,

59

Since multiple valid values of

N

exist, Statement (II) alone is NOT sufficient.

Evaluate Statement (I) and Statement (II) together.

System of linear equations:

x + y = 12

and

y - x = 4

. Adding the two equations gives

2y = 16 \implies y = 8

. Substituting

y = 8

gives

x = 4

. Thus,

N = 48

uniquely.

Both statements combined provide a unique value for

N

, making them together sufficient.

Key Concept

Data Sufficiency in Digit-Based Integer Problems

Estimated Time:1m 30s

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