HCF and LCM Questions

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.

What is HCF?

HCF stands for Highest Common factor. HCF for two or more numbers is the highest number that divides all the values with the remainder being zero. HCF is also commonly known as GCD (Greatest Common Divisor).

For example, the highest Common factor for 6 and 24 is 6. As 6 is the highest common value by which between both 6 and 24 are divisible with the remainder being zero.

HCF of 6 and 24

What is LCM?

LCM stands for Lowest Common Multiple. LCM of two or more numbers is the smallest number that is the multiple of all the given numbers. We can also say that the LCM of given numbers are the smallest number which is exactly divisible by the given numbers.

For example the Lowest Common Multiple od 3 and 7 is 21. As 21 is the smallest number that is a mutiple of both 3 and 7.

LCM of 3 and 7

How to calculate HCF and LCM?

There are two famous methods to calculate both HCF and LCM:

• Prime factorization Method: Prime factorization is a mathematical method used to express a composite number as the product of its prime factors. Example 100 = 2 x 2 x 5 x 5.
• Division Method: This method is also known as long division method. It is a technique used to determine both the quotient and remainder in the process of dividing one number (known as the dividend) by another (referred to as the divisor). This step-by-step procedure hinges on the principles of repeated subtraction and multiplication. For example when 3 divides 16 it gives 5 as quotient and 1 as remainder.

Calculation for HCF

HCF can be calculated using two methods. These are Prime Factorization Method and Division method. These two methods are discussed below:

HCF by Prime Factorisation Method

As the name suggests, in this method we first find the prime factors of the given numbers and using the prime factors of the given numbers we find the HCF. Let’s understand it with the help of an example.

Example: Find the HCF 16 and 72 using prime factorisation method.

Solution:

Step 1: find the prime factors of all the given numbers.

16 = 2 x 2 x 2 x 2

72 = 2 x 2 x 2 x 3 x 3

Step 2: Check the common factors between them.

Common factors between 16 and 72 = 2 x 2 x 2

Step 3: Multiply the common values to get the final answer

HCF ( 16, 72) = 2 x 2 x 2 = 8

HCF by Division Method

In this method we divide the larger number by the smaller number out of the two given number of which we have to find the HCF. We successively divide the previous divisior by the obtained remainder. This process continues till the remainder isn zero. Let’s understand it with an example.

Example: Find the HCF of 12 and 18 by Division Method.

Solution:

Step 1: Begin by writing down the numbers you want to find the HCF for. For this example, let’s use 12 and 18.

Step 2: Divide 18 by 12 i.e. 18 ÷ 12, Q₁ = 1, R₁ = 6

Step 3: Divide the previous divisor i.e. 12 by the remainder i.e. 6. This means now 12 is the dividend and 6 is the divisior. We get Q₂ = 2 and R₂ = 0

Step 4: Since, the Remainder is zero when the divisor is 6. Hence HCF is 6

Step 5: If the remainder would not have been zero then we would continue the process till we get the remainder to be zero. The divisor of the last step is the HCF of the numbers.

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.

Calculation for LCM

LCM can be calculated using two methods. These are Prime Factorization Method and Division method. These two methods are discussed below:

LCM by Prime Factorisation Method

As the name suggests, in this method we first find the prime factors of the given numbers and using the prime factors of the given numbers we find the LCM. Let’s understand it with the help of an example.

Example: Find the LCM 16 and 72 using prime factorisation method.

Solution:

Step 1: find the prime factors of all the given numbers. 16 = 2 x 2 x 2 x 2 72 = 2 x 2 x 2 x 3 x 3

Step 2: Check the count of different factors of the given values. 2 : Four times in (16) 2 : three times in (72) 3 : two times in (72)

Step 3: Now the common factor based on the higher count will be taken and the rest will be ignored. 2: Four times (16) > 2: three times (72) higher count will taken. 3 is only present in the (72) hence it will be taken.

Step 4: Multiply the selected values to get the final answer LCM ( 16, 72) = 2 x 2 x 2 x 2 x 3 x 3 = 144.

LCM by Division Method

We can find LCM of two numbers using the division method. In this method we divide the given numbers by their common factors and then successively divide the quotient to get another quotient. This process continues till we get the quotient such that they are not divisible by any common number. Let’s understand this with the help of an example

Example: Find the LCM of 24 and 30 by Division Method.

Solution:

Step 1: Start by listing the numbers for which you want to find the LCM. For this we have 24 and 30.

Step 2: Identify the smallest prime number that can evenly divide both 24 and 30. In this case, it’s 2. Write down 2 on the left.

Step 3: Divide both 24 and 30 by 2. Write the quotients below the numbers. For 24 ÷ 2, the quotient is 12, and for 30 ÷ 2, the quotient is 15.

Step 4: Keep looking for common prime factors. If there are more, repeat the process. For 12 and 15, the common factor is 3. Write down 3 on the left.

Step 5: Divide 12 and 15 by 3. Write the quotients below the numbers. For 12 ÷ 3, the quotient is 4, and for 15 ÷ 3, the quotient is 5.

Step 6: Once there are no more common prime factors, the numbers inside the green box are the prime factors of the LCM. To calculate the LCM, multiply all these numbers together.

Step 7: So, LCM of 24 and 30 = 2 × 3 × 4 × 5 = 120.

LCM Questions with Solution

Question 1: Find the LCM of 9, 12 and 15.

Answer:

Using Division Method,

LCM(9, 12 , 15) = 2 x 2 x 3 x 3 x 5 = 180

Question 2: Find the LCM of 8, 12 and 18.

Answer:

Using Prime Factorisation Method,

8 = 2 x 2 x 2

12 = 2 x 2 x3

18 = 2 x 3 x 3

Remember always consider prime factors that has the maximum count.

LCM(8, 12, 18) = 2 x 2 x 2 x 3 x 3 = 72

Question 3: Can the LCM of two numbers be smaller than both numbers?

Answer:

No, the LCM of two numbers cannot be smaller than both of those numbers. The LCM is defined as the smallest multiple that both numbers share in common.

Mathematically, if you have two numbers, let’s say ‘a’ and ‘b,’ where ‘a’ and ‘b’ are positive integers, then:

LCM(a, b) ≥ a and LCM(a, b) ≥ b

Hence LCM is always greater than or equal to both of the original numbers.

Question 4: Can the LCM of two numbers be zero?

Answer:

No, the LCM of two non-zero numbers cannot be zero. The LCM is defined as the smallest positive multiple that two numbers have in common. Since it’s the smallest multiple, it must be greater than or equal to 1.

Mathematically, if you have two non-zero numbers, ‘a’ and ‘b,’ where ‘a’ and ‘b’ are positive integers:

LCM(a, b) > 0

This means that the LCM is always a positive integer and cannot be zero, regardless of the values of ‘a’ and ‘b.’ Zero is not a valid LCM because it does not represent a common multiple of two non-zero numbers.

Question 5: Sarah is making friendship bracelets. She wants to make bracelets that are 8 inches long and 12 inches long. What is the smallest length of string she can use for each bracelet if she doesn’t want to have any leftover string?

Answer:

To find the smallest length of string that Sarah can use for each bracelet without any leftover string, we need to calculate the LCM of 8 inches and 12 inches.

Prime Factorisation Method ,

8 = 2 x 2 x 2

12 = 2 x 2 x 3

LCM(8, 12) = 2 x 2 x 2 x3 = 24

The smallest number that appears in both lists is 24. Hence Sarah needs a string that is 24 inches long for each bracelet to avoid any leftover string.

Question 6: Emily is a gardener, and she wants to plant flowers in her garden in rows. She has two types of flowers: roses and tulips. Emily wants to plant her flowers in rows such that each row contains the same number of each type of flower. She has 6 rose plants and 8 tulip plants. What is the maximum number of plants Emily can put in each row so that no plants are left over?

Answer:

To find the maximum number of plants Emily can put in each row so that no plants are left over, we need to calculate the LCM of the number of rose plants and the number of tulip plants.

Using Division Method,

LCM(6, 8) = 2 x 3 x 4 =24

The smallest number that appears in both lists is 24. Hence, Emily can plant 24 plants in each row, with 12 roses and 12 tulips, so that no plants are left over.

Relation Between HCF and LCM

The HCF of two numbers say, A and B multiplied by the LCM of A and B are always going to be equal to the product of A and B.

HCF (A,B) x LCM (A,B) = A x B

Example: Prove that HCF(4,3) x LCM(4,3) = 4×3

Solution:

HCF (4,3) = 1

LCM (4,3) = 12

LHS: 12 ⨯ 1 = 12

RHS: 4 ⨯ 3 = 12

LHS = RHS

Hence proved.

Article Related to HCF & LCM Questions: HCF & LCM

HCF and LCM Tricks

HCF (Highest Common Factor):

1. The HCF of numbers is never greater than any of the numbers. It’s always smaller.
2. When you have prime numbers, the HCF is always 1. Example 3 and 2 HCF(3,5) = 1.

LCM (Least Common Multiple):

1. The LCM of numbers is never less than any of the numbers. It’s always equal to or greater.
2. When you have prime numbers, the LCM is just the result of multiplying them. Example 3 and 2 LCM(3,5) = 15.

HCF and LCM for Fractions:

1. To find the LCM of fractions, multiply the LCM of the numerators and divide by the HCF of the denominators.
2. To find the HCF of fractions, multiply the HCF of the numerators and divide by the LCM of the denominators.

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 and LCM Problems

Q1. If the LCM of three numbers is 180 and their HCF is 6, what are the numbers?

Q2. the HCF of two numbers is 8, and their product is 64. What are the numbers?

Q3. Find the HCF and LCM of (91/37) and (15/64)?

Q4. What smallest value when divided by 36, 24 and 16 leaves remainder in each case?

Q5. Find the smallest value which when divided by 12 leaves a remainder of 10 when divided by 24 leaves a remainder of 22 and when divided by 32 leaves a remainder of 30.

Q6. Find the highest value that will divide 43, 91 and 183 so as to leave the same remainder in each case.

HCF and LCM – FAQs

1. What is HCF?

HCF is the Highest Common factor of a set of numbers is the highest value that divides two or more number without leaving any remainder or remainder = 0.

2. What is LCM?

LCM is the Lowest Common Multiple of a set of numbers is lowest multiple that they have in common

3. What is GCD?

GCD stands for Greatest Common Divisor and it is another term that refers to HCF.

4. How can we calculate HCF or LCM of more than two values?

You can first calculate the HCF or LCM of any two values and than you can use the result to calulate with the third value and so on for n number of value.

5. What are Co-Primes?

Two values are called co-prime when there HCF is equal to 1. Example 9 and 8.