A numerical analysis task requires evaluating four specific values to classify them into t… — State PSC Exam

A numerical analysis task requires evaluating four specific values to classify them into their correct number sets. The values are defined as follows:

- $K = 18 - 6 \div 2 + 1$
- $L =$ The unit digit of $8^{32}$
- $M =$ The remainder when $-23$ is divided by $5$
- $N =$ The Highest Common Factor (HCF) of $\frac{3}{4}$ and $\frac{9}{10}$

Based on the correct mathematical evaluation of these expressions, which of the following statements regarding their classification are mathematically correct?

The value of $K$ is a perfect square, and $M$ is the only prime number among the integer results.Answer
The value of $N$ is a rational fraction strictly between $0$ and $1$ , and $L$ is an even composite number.Answer
C
The value of $K$ is a prime number, and $N$ is an improper fraction greater than $1$ .
D
The value of $L$ is a positive odd integer, and $M$ is a negative integer.

Answer

The mathematically correct statements are that K is a perfect square, M is the only prime among the integer results, N is a rational fraction between 0 and 1, and L is an even composite number.

Based on rigorous mathematical evaluation, the true values are

K = 16

L = 6

M = 2

, and

N = \frac{3}{20}

K

(

16

) is a perfect square, and

M

(

2

) is the only prime among the integer results (

16, 6, 2

). Furthermore,

N

(

0.15

) is a rational fraction strictly between

0

and

1

, and

L

(

6

) is an even composite number. Therefore, these descriptive classifications perfectly match the evaluated properties.

Step-by-Step Solution

Evaluate expression K using proper BODMAS rules.

K = 16

Division must be performed before addition and subtraction.

K = 18 - (6 \div 2) + 1 = 18 - 3 + 1 = 16

, which classifies as a perfect square.

Determine the unit digit of L based on cyclicity.

L = 6

The unit digit of powers of

8

follows a 4-step cycle (

8, 4, 2, 6

). Since the exponent

32

is perfectly divisible by

4

, the unit digit is the 4th in the cycle, which is

6

(an even composite number).

Calculate the mathematically correct positive remainder for M.

M = 2

By the formal division algorithm,

-23 = 5 \times (-5) + 2

. Remainder must be non-negative, so the remainder is

2

, which is an even prime number.

Compute the HCF of the given fractions for N.

N = \frac{3}{20}

The HCF of fractions is computed as

\frac{\text{HCF of numerators}}{\text{LCM of denominators}}

\text{HCF}(3,9) = 3

and

\text{LCM}(4,10) = 20

. Thus

N = \frac{3}{20} = 0.15

, a rational number strictly between

0

and

1

Cross-reference the correctly evaluated numbers against the provided statements.

The statements categorizing

K

as a perfect square,

M

as the only prime among integers,

N

between

0

and

1

, and

L

as an even composite are correct.

The integer results are

16, 6

, and

2

, where exactly one (

2

) is prime. The other statements rely on distinct computational and conceptual errors.

Key Concept

Applying fundamental arithmetic rules and modular arithmetic to properly classify numbers into distinct mathematical sets.

Estimated Time:1m 30s

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