NCERT Solutions Class 7 Maths Chapter 11 — HCF & LCM

Solution:

Step 1: Factors are limited, meaning a number has a finite set of factors. Multiples are infinite. So, option A is false.

Step 2: The smallest multiple of any natural number (except 0) is the number itself. All other multiples are greater than the number. So, option B is true.

Step 3: The only common factor of two prime numbers is 1, as prime numbers only have 1 and themselves as factors. So, option C is false.

Step 4: The smallest multiple of every number is the number itself, not 1 (unless the number is 1). So, option D is false.

Solution:

Step 1: Rohan's final product (2 × 3 × 2 × 2 × 3 = 72) is numerically correct, meaning the prime factors and their counts are right.

Step 2: However, the *process* shown has a conceptual flaw. In the step '36 = 3 × 12', he broke 36 into a prime (3) and a composite (12). Similarly, '12 = 2 × 6' involves a composite (6).

Step 3: A proper prime factorization method involves breaking down *all* composite factors into prime factors at each step, ensuring that at any point, all factors listed are either prime or are immediately broken down until only primes remain.

Step 4: Thus, the error is in not consistently breaking down composite numbers (like 36, 12, and 6) into *only* prime factors at each stage of the decomposition before moving on, making the process less systematic.

Solution:

Step 1: Option A correctly states that HCF is the largest number that divides each given number without leaving a remainder. This is the fundamental definition of HCF.

Step 2: Option B describes the Least Common Multiple (LCM), not HCF.

Step 3: Option C describes a method to calculate HCF using prime factorization, but it's not the definition of HCF itself as a property.

Step 4: Option D is incorrect; HCF doesn't have to be prime, and it must divide *all* given numbers, not just at least one.

Solution:

Step 1: Option A describes the HCF, not LCM.

Step 2: Option B correctly states that LCM is the smallest positive number that is a multiple of all the given numbers. This is the fundamental definition of LCM.

Step 3: Option C describes a method to calculate LCM using prime factorization, but it's not the definition of LCM itself as a property.

Step 4: Option D is incorrect; LCM doesn't have to be prime, and it's a multiple, not a divisor.

Solution:

Step 1: First, find the prime factorization of each number: [48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹]

Step 2: Next, find the prime factorization of the second number: [72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²]

Step 3: To find the HCF, take the common prime factors (2 and 3) with the lowest power they appear in either factorization. [Lowest power of 2 = 2³ Lowest power of 3 = 3¹]

Step 4: Multiply these lowest powers together to get the HCF: [HCF(48, 72) = 2³ × 3¹ = 8 × 3 = 24]