HCF Questions with Solutions

Question 1: Find the HCF of 36 and 48.

Answer:

Using Prime Factorisation Method

36 = 2 x 2 x 3 x 3

48 = 2 x 2 x 2 x 2 x 3

check for common factors between them

HCF(36,48) = 2 x 2 x 3 = 12

Question 2: Calculate the HCF of 16, 24 and 40.

Answer:

Using Division Method

In this method we divide the numbers with a common factor till the quotient obtained have no common factor and they we multiply the divisors to obtain the HCF.

HCF(16,24,40) = 2 x 2 x 2 = 8

Question 3: Find the greatest number that will divide 12, 36 and 54 exactly?

Answer:

Using Prime factorization method

12 = 2 x 2 x 3

36 = 2 x 2 x 3 x 3

54 = 2 x 3 x 3 x 3

HCF (12, 36, 54) = 2 x 3 =6

Question 4: If the HCF of two numbers is 1, what can you conclude about those numbers?

Answer:

When the HCF of two numbers is 1, it indicates that these two numbers are coprime . As Coprime numbers are integers that share no common factors other than 1. In other words, their HCF is the smallest possible, which is 1. This implies that the numbers have no common factors except for unity, making them mutually prime to each other. For example, 5 and 8 are coprime because their only common factor is 1.

Question 5: Can the HCF of two prime numbers be a prime number other than 1? Explain.

Answer:

The HCF (Highest Common Factor) of two prime numbers cannot be a prime number other than 1.

Consider the scenario where the HCF of two numbers is a prime number other than 1. Let’s say the HCF is ‘p’, where ‘p’ is a prime number greater than 1.

• If ‘p’ is the HCF, it means ‘p’ is a common factor of the two numbers.
• However, since ‘p’ is a prime number greater than 1, it cannot be divided by any other number except 1 and itself.
• This implies that the two numbers can only be divided by ‘p’ and nothing else.
• But this contradicts the definition of HCF because the HCF is supposed to be the largest number that can exactly divide both of them. If ‘p’ is the HCF, it should be the largest, but it cannot be because it cannot be divided by any number other than 1 and itself.

Therefore, the HCF of two numbers cannot be a prime number other than 1. It must always be 1 or a composite number (a number with more than two factors).

Question 6: Sarah has 12 apples, and she wants to arrange them into equal-sized groups. She wants to ensure that no apples are left over in each group. What is the largest number of apples she can put in each group?

Answer:

To find the largest number of apples that Sarah can put in each group without any apples left over, we need to calculate the HCF of the number of apples she has, which is 12.

The HCF of 12 is 12 itself, so Sarah can put 12 apples in each group, and no apples will be left over.

HCF (Highest Common Factor) and LCM (Least Common Multiple) are fundamental concepts in mathematics, particularly in number theory. HCF is the highest common number which can exactly divide the two given numbers. LCM or Lowest Common Multiple is the common number that is divisible by both the given numbers. These concepts are essential tools for solving a wide range of mathematical problems.

In this article, we will learn about the definitions of HCF and LCM, their properties, and methods for calculating HCF and LCM. Along with this, all the possible varieties of HCF and LCM Questions have been discussed with solutions, and practice questions are provided on HCF and LCM for learners.