Analyze the given mathematical claims regarding the classification and properties of numbe… — State PSC Exam

Analyze the given mathematical claims regarding the classification and properties of numbers:

I. If $p$ is a prime number strictly greater than $3$ , then $p^2 - 1$ is always a composite number divisible by $24$ .
II. The least common multiple (LCM) of the rational numbers $\frac{6}{5}$ and $\frac{12}{7}$ is a composite integer, while their highest common factor (HCF) is a non-integer rational number.
III. The numerical value of the expression $36 \div 6 \times 3$ belongs to the set of prime numbers.
IV. According to standard Euclidean division, the remainder when $-22$ is divided by $7$ is $-1$ , which is classified as a negative integer.

Which of the above claims are mathematically correct?

A
Only I, II, and III
Only I and IIAnswer
C
Only I, II, and IV
D
Only I and IV

Answer

The only mathematically correct claims are I and II.

The correct answer accurately identifies that only Claims I and II are valid. Claim I holds because the square of any prime

p>3

is congruent to

1 \pmod{24}

, making

p^2 - 1

a composite multiple of 24. Claim II is valid as the LCM of

6/5

and

12/7

evaluates precisely to the composite integer 12, and their HCF evaluates to

6/35

. Claims III and IV are mathematically unsound.

Step-by-Step Solution

Evaluate Claim I by testing the property of primes greater than 3.

Claim I is TRUE.

Any prime

p > 3

can be expressed as

6k \pm 1

. Thus,

p^2 - 1 = (6k \pm 1)^2 - 1 = 36k^2 \pm 12k = 12k(3k \pm 1)

. Since either

k

3k \pm 1

must be even, the expression is always a multiple of

12 \times 2 = 24

. Since

p \ge 5

p^2 - 1 \ge 24

, so it is always a composite number divisible by 24.

Evaluate Claim II by calculating the LCM and HCF of fractions

\frac{6}{5}

and

\frac{12}{7}

Claim II is TRUE.

LCM of fractions = LCM(numerators)/HCF(denominators) = LCM(6,12)/HCF(5,7) = 12/1 = 12, which is a composite integer. HCF of fractions = HCF(numerators)/LCM(denominators) = HCF(6,12)/LCM(5,7) = 6/35, which is a non-integer rational number.

Evaluate Claim III by applying the correct order of operations (BODMAS) to

36 \div 6 \times 3

Claim III is FALSE.

Division and multiplication have equal precedence and are evaluated from left to right.

36 \div 6 = 6

, and

6 \times 3 = 18

. The number 18 is composite, not prime.

Evaluate Claim IV by determining the correct Euclidean remainder of

-22 \div 7

Claim IV is FALSE.

In standard Euclidean division, the remainder

r

must satisfy

0 \le r < |divisor|

. For

-22

divided by

7

, the correct equation is

-22 = 7 \times (-4) + 6

. The remainder is 6 (a positive integer), not

-1

Key Concept

Classification of Numbers and Number Properties

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