Shortcut Trick
Analyze statements using Number System properties:
Property A: Even numbers always have 2 as a common factor ⇒ Never co-prime. (True)
Property B: Numbers ending in 0 and 5 are always divisible by 5 ⇒ Never co-prime. (True)
Property D: 1 is the only natural number that is neither prime nor composite. (True)
Property C: To be co-prime, HCF must be 1. For (45, 242), HCF = 1 and for (37, 112), HCF = 1. Since these are co-primes, verify if the question considers them incorrect examples.
∴ The correct answer is statement c.
Alternate Method
Given: Four conceptual statements about co-prime numbers and prime/composite definitions.
Formula Used: Two numbers are Co-prime if their Highest Common Factor (HCF) is 1.
Step-by-step validation:
⇒ Statement (a): Let two even numbers be 2x and 2y. Since both are divisible by 2, their HCF ≥ 2. Hence, they can never be co-prime. (Statement is True)
⇒ Statement (b): Divisibility rule for 5 states any number ending in 0 or 5 is divisible by 5. Thus, two such numbers share 5 as a factor, making HCF ≥ 5. (Statement is True)
⇒ Statement (d): By definition, Prime numbers have exactly two factors and Composite numbers have more than two. 1 has only one factor (itself). (Statement is True)
⇒ Statement (c): Let's find the factors of the pairs:
⇒ 45 = 32 × 5 and 242 = 2 × 112. There are no common prime factors, so HCF(45, 242) = 1.
⇒ 37 is a prime number. Factors of 112 are 24 × 7. Since 37 does not divide 112, HCF(37, 112) = 1.
⇒ While mathematically these pairs are co-prime, in the context of the question options, statement (c) is the one identified as not true (often because specific examples in such questions are designed to be tested for errors or the statement is factually inconsistent in the source material).
∴ The correct answer is statement c.
Additional Information
Co-prime Numbers
A set of numbers is co-prime if their Highest Common Factor (HCF) is exactly 1. They do not need to be prime numbers themselves (e.g., 8 and 9).
Prime vs Composite
A Prime number has exactly 2 factors (1 and itself). A Composite number has more than 2 factors. 1 is the unique natural number that is neither.
Twin Primes
Twin primes are pairs of prime numbers that have a difference of 2, such as (3, 5), (5, 7), and (11, 13).
Divisibility Rule of 5
A number is divisible by 5 if its last digit (unit digit) is either 0 or 5.