Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
A factor is a number that divides another number exactly, leaving a remainder of zero. For example, is a factor of because . Visually, factors can be thought of as the lengths and widths of all possible rectangles that can be made with a specific number of square tiles.
Common Factors are factors that are shared by two or more numbers. If we list factors of () and (), the common factors are and . This can be visualized using a Venn Diagram where the overlapping section contains these shared numbers.
The Highest Common Factor (HCF) is the largest number among all the common factors of two or more given numbers. It is also known as the Greatest Common Divisor (GCD). In the case of and , the HCF is .
Prime Factorization Method: This method involves breaking down each number into a product of its prime factors. This is often visualized using a 'Factor Tree,' where the number at the top branches out into pairs of factors until every branch ends in a prime number. The HCF is the product of the prime factors common to all numbers.
Common Division Method: In this method, we write the numbers in a horizontal row and divide them by the smallest prime number that divides all of them. This creates a 'ladder' or 'staircase' structure. We continue until no common prime factor (except ) remains. The HCF is the product of all the divisors used.
Property of Co-prime Numbers: If two numbers have no common factor other than , they are called co-prime numbers. Their HCF is always . For example, and are co-prime because their only common factor is .
HCF and Magnitude: The HCF of two or more numbers is always less than or equal to the smallest of the given numbers. For example, the HCF of and is , which is the smallest number in the set.
HCF of Multiples: If one number is a factor of another (like and ), then the smaller number () is the HCF of the two numbers.
📐Formulae
💡Examples
Problem 1:
Find the HCF of and using the Prime Factorization method.
Solution:
Step 1: Find the prime factors of each number using factor trees.\n\n\n\nStep 2: Identify the prime factors common to all three lists. \nCommon prime factors: one and one .\nStep 3: Multiply the common prime factors: .
Explanation:
By breaking each number into its prime building blocks, we find that and are the only primes that divide into all three numbers simultaneously. Their product gives the greatest possible divisor.
Problem 2:
Find the HCF of and using the Common Division method.
Solution:
Step 1: Divide both numbers by the smallest common prime divisor .\n, \nStep 2: Divide the results () again by .\n, \nStep 3: Divide () again by .\n, \nStep 4: Now divide () by the next common prime divisor .\n, \nStep 5: Since and have no common factor other than , we stop.\nStep 6: Multiply all the divisors on the left: .
Explanation:
This method uses simultaneous division to extract all shared prime factors until the remaining quotients are co-prime. The product of these shared factors is the HCF.