HCF & LCM MCQ Test — Class 7 CBSE

Solution:

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

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.

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.

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

Solution:

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

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).

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.

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:

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.

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

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

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

Solution:

Option A describes the HCF, not LCM.

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.

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

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

Solution:

First, find the prime factorization of each number: — 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹

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

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

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

Solution:

First, find the prime factorization of each number: — 12 = 2 × 2 × 3 = 2² × 3¹

Next, for the other numbers: — 18 = 2 × 3 × 3 = 2¹ × 3² 24 = 2 × 2 × 2 × 3 = 2³ × 3¹

To find the LCM, take all unique prime factors (2 and 3) with the highest power they appear in any of the factorizations. — Highest power of 2 = 2³ Highest power of 3 = 3²

Multiply these highest powers together to get the LCM: — LCM(12, 18, 24) = 2³ × 3² = 8 × 9 = 72

Solution:

Co-prime numbers (or relatively prime numbers) are defined as two integers that have no common factors other than 1.

By definition, the Highest Common Factor (HCF) of co-prime numbers must therefore be 1.

For example, the numbers 7 and 10 are co-prime. The factors of 7 are {1, 7} and the factors of 10 are {1, 2, 5, 10}. The only common factor is 1, so HCF(7, 10) = 1.

Solution:

If the HCF of two numbers is 6, it means that both numbers must be divisible by 6 (i.e., they must be multiples of 6).

Let's check each option to see which number is NOT a multiple of 6:

A) 18: 18 ÷ 6 = 3 (18 is a multiple of 6). So, 18 could be the other number (HCF(30, 18) = 6).

B) 42: 42 ÷ 6 = 7 (42 is a multiple of 6). So, 42 could be the other number (HCF(30, 42) = 6).

C) 54: 54 ÷ 6 = 9 (54 is a multiple of 6). So, 54 could be the other number (HCF(30, 54) = 6).

D) 40: 40 is not divisible by 6 (40 ÷ 6 leaves a remainder). Therefore, 40 cannot have 6 as its HCF with any number.

Solution:

To find when the bells will ring together again, we need to find the Least Common Multiple (LCM) of their ringing intervals (15 minutes and 20 minutes).

Find the prime factorization of each number: — 15 = 3 × 5 20 = 2² × 5

Calculate the LCM by taking all unique prime factors with their highest powers: — LCM(15, 20) = 2² × 3 × 5 = 4 × 3 × 5 = 60 minutes

Since 60 minutes is equal to 1 hour, the bells will ring together again 1 hour after 10:00 AM. So, they will next ring together at 11:00 AM.

Solution:

The phrase 'greatest number that can divide 42 and 63 exactly' asks for the largest common factor of 42 and 63. This is the definition of the Highest Common Factor (HCF).

The phrase 'smallest number that is exactly divisible by 12 and 18' asks for the smallest number that is a common multiple of 12 and 18. This is the definition of the Least Common Multiple (LCM).

Therefore, the blanks should be filled with HCF and LCM respectively.