Solved Questions on HCF and LCM

Example 1: If the LCM of two numbers is 72 and one of the numbers is 18, what is the other number?

Solution:

Let the two numbers be a, b.

Given :

LCM(a,b) = 72

let a =18.

We can use the equation ,

HCF x LCM = product of two numbers

HCF(a,b) x LCM(a,b) = a x b;

To find HCF(18, b), you can use the fact that 18 is a factor of 72 (18 * 4 = 72), so HCF(18, b) = 18.

b = (HCF(a,b) x LCM(a,b))/a

b = (72 * 18)/18

b = 72, hence the other number is 72

Example 2: Find the HCF of (15/30 ) and (20/40).

Solution:

HCF of fractions = HCF(numerator)/LCM(denomiantor)

HCF of fractions = HCF(15,20)/LCM(30,40)

HCF of fractions = 5/120

Example 3: Find the LCM of (1/5) and (2/3).

Solution:

LCM of fractions = LCM(numerator)/HCF(denominator)

LCM of fractions = LCM(1,2)/HCF(5,3)

LCM of fractions = 2/1 = 2

Example 4: Sarah is preparing dinner plates. She has 60 pieces of momos and 8 rolls. If she wants to make all the plates identical without any food left over, what is the greatest number of plates Sarah can prepare ?

Solution:

In order for all the plates to look identical with the highest no of plates, we need to find the hcf of 8 rolls and 60 pieces of momos

HCF (8,60) = 4

Hence Sarah can prepare a total of 4 plates

Example 5: A juice seller has three different types of fruit juices: apple juice, orange juice, and grape juice. He has 403 liters of apple juice, 434 liters of orange juice, and 465 liters of grape juice. What is the minimum number of identical containers he needs to store each type of juice separately without mixing them?

Solution:

For the minimum number of containers of equal size, the size of each container must be of the greatest volume.

To get the greatest volume of each container, we need to find HCF of 403, 434 and 465.

H.C.F (403, 434, 465) = 31 liters

Each container must be of the volume 31 liters.

Number of containers required are = (403/31) + (434/31) + (465/31) = 42

Hence, the minimum number of containers required are 42.

Example 6: Determine the minimum number of students needed to form perfect square groups for a school activity where they stand in rows of 15, 20, and 25 students each.

Solution:

To find the minimum number of students needed for the school activity to stand in rows of 15, 20, and 25 while forming a perfect square, we must first calculate the least common multiple (LCM) of these numbers.

LCM(15, 20, 25) = 300

So, ideally, we would need 300 students to meet the divisibility criteria. However, the problem statements states that this number should be a perfect square.

To achieve this, we can multiply 300 by 3, resulting in 900. Now, 900 is a perfect square (30 × 30).

Hence, the minimum number of students required to form rows of 15, 20, and 25, while also forming a perfect square, is 900.

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.