How to Solve HCF Questions Using Prime Factorization and Division Method
Highest Common Factor (HCF) as the name itself says, it's the method of finding the highest common factors of two or more than two numbers. It's the highest common number which can divide the given two or more than two positive numbers equally.
To find the HCF, we first need to find out the factors of each given number. Then we check for the common factors where the factors that we found out are the same and lastly we mark the highest factor of them all.
For example:
Let's take two positive numbers
1. Find out the HCF of 6,12.
Ans:
Step 1- Finding out the factors
Factors of 6,12
Step 2: finding out the common factors
The factors of 6 and 12 are 1, 2, 3, 6
The Highest Common Factor of (6,12) = 6
Understanding How to Find HCF Using Short Division Method
1. Find out the HCF of 36 and 45.
Ans: The easiest method of finding out the HCF of large numbers is using the short division method.
For this method, we need to find out the LCM (least Common Multiple) of 36 and 45. Firstly draw a table and arrange the given numbers horizontally. We need to find out the common factors which can divide the given number equally and put it on the left hand side.
LCM Method to Find Out Common Factors
We keep on dividing until we get co-prime numbers at the end. Prime number is a number that cannot be exactly divided by any other whole number other than itself and 1.
In this case. we got 4 and 5 which don't have common factors. Lastly, we multiply all the common factors that we have.
3X3 = 9
Therefore, the HCF of (36, 45) = 9.
HCF Word Problem
1. A milk seller has 120 liters of almond milk and 180 liters of soya milk. He wants to sell the milk by filling both the kinds of milk in tins of equal amount. Find the greatest volume of which the tin can take.
Ans: To find out the equal quantity of both the milks that the tin can take, we need to find out the factors of 120 and 180;
Well the target of the question is to find out the greatest volume of milks filled in the tins. So, we have to find the largest number which exactly divides 120 and 180 equally which means we will have to find out the HCF of 120 and 180.
Using LCM to find out the common factors
LCM of 120, 180
HCF of (120, 180)=> 2X2X15 = 60
Therefore, the greatest volume that the tins can take is 60 liters of each.
Which means 120 ÷ 60 = 2 tins of almond milk
180 ÷ 60 = 3 tins of soya milk.
Practice Questions
1. Sneha has 40 strawberries and Eisha has 95 blueberries. They want to distribute the fruits in such a way that each box has an equal number of fruits and also each box should have the exact set of strawberries or blueberries. What is the greatest number of fruits that can be arranged in each box?
Ans: HCF of (40,95)= 5
Therefore, the fruits in the boxes can be arranged equally in the set of 5.
2. Two wooden boards are 77 m and 44 m long respectively. The wooden boards are to be cut into pieces of equal length. Find the maximum length of each board.
Ans: HCF (77,43)= 11
Therefore, the wooden boards can be cut into 11 equal lengths.
Summary
In this article we learnt that HCF (highest common factor) also known as GCD (greatest common divisor) basically is the number which can divide the given two or more numbers equally and one of the easiest ways of finding out the HCF of large numbers is using the Short division method. The article tries to explain the short division method step-by-step properly.
FAQs on HCF Questions and Problems Explained with Methods
1. What is HCF in Maths?
The Highest Common Factor (HCF) is the greatest number that divides two or more numbers exactly without leaving a remainder. It is also called the Greatest Common Divisor (GCD).
- It is always less than or equal to the smallest given number.
- It must divide all the given numbers completely.
- Example: The HCF of 12 and 18 is 6 because 6 is the largest number that divides both exactly.
2. How do you find the HCF of two numbers?
You can find the HCF of two numbers using the listing method, prime factorization method, or division method. The most common method is prime factorization.
- Step 1: Write the prime factors of each number.
- Step 2: Identify common prime factors.
- Step 3: Multiply the common prime factors.
- Example: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3.
Common factors: 2 × 2 × 3 = 12.
So, HCF = 12.
3. What is the formula for HCF?
The formula relating HCF and LCM of two numbers is: HCF × LCM = Product of the two numbers.
- If two numbers are a and b, then:
- HCF(a, b) × LCM(a, b) = a × b
- Example: For 8 and 12, HCF = 4 and LCM = 24.
4 × 24 = 96 and 8 × 12 = 96.
4. What is the difference between HCF and LCM?
The HCF is the greatest common divisor of given numbers, while the LCM is the smallest common multiple of those numbers.
- HCF deals with divisibility.
- LCM deals with multiples.
- Example: For 6 and 8, HCF = 2 and LCM = 24.
5. How do you find the HCF using the division method?
The division method finds HCF by repeatedly dividing until the remainder becomes zero. The last non-zero remainder is the HCF.
- Step 1: Divide the larger number by the smaller number.
- Step 2: Replace the larger number with the smaller number and the smaller number with the remainder.
- Step 3: Repeat until remainder = 0.
- Example: 48 ÷ 18 → remainder 12; 18 ÷ 12 → remainder 6; 12 ÷ 6 → remainder 0.
HCF = 6.
6. Can you give an example of an HCF word problem?
An HCF word problem usually asks for the greatest possible size or grouping that divides quantities equally.
- Example: 24 apples and 36 oranges are to be packed into boxes with equal numbers in each box.
- HCF of 24 and 36 = 12.
- So, 12 boxes can be made, each containing 2 apples and 3 oranges.
7. What is the HCF of co-prime numbers?
The HCF of two co-prime numbers is always 1 because they have no common factors other than 1.
- Example: 8 and 15 have no common prime factors.
- So, HCF(8, 15) = 1.
8. How do you find the HCF of three numbers?
To find the HCF of three numbers, first find the HCF of any two numbers, then find the HCF of the result with the third number.
- Example: Find HCF of 12, 18, and 24.
- HCF(12, 18) = 6.
- HCF(6, 24) = 6.
- So, HCF of 12, 18, 24 = 6.
9. Why is HCF important in Maths?
The HCF is important because it helps simplify fractions, solve word problems, and divide quantities into equal groups.
- Used in reducing fractions to lowest terms.
- Helps in solving real-life grouping and distribution problems.
- Forms the basis for understanding LCM and number theory concepts.
10. What are common mistakes while solving HCF problems?
Common mistakes in HCF problems include confusing HCF with LCM and missing common prime factors.
- Multiplying all factors instead of only common ones.
- Stopping the division method too early.
- Ignoring repeated prime factors in prime factorization.
- Always check that the final HCF divides all given numbers exactly.
