Introduction to Finding the HCF of 525 and 289
When working with numbers, finding the highest common factor (HCF) is an important concept in mathematics. This guide will walk you through a detailed explanation of how to find the HCF of the numbers 525 and 289 using the Euclidean algorithm. We'll start by examining the prime factorization of each number and then apply the Euclidean algorithm step-by-step.
Prime Factorization of 525 and 289
To start, let's break down each number into its prime factors.
Prime Factorization of 525
525 can be factored as:
525 3 × 5 × 5 × 7
Prime Factorization of 289
289 can be factored as:
289 17 × 17
From the prime factorizations, it is clear that these two numbers do not share any prime factors except for 1. Therefore, the HCF is 1.
Applying the Euclidean Algorithm to Find HCF
The Euclidean algorithm is a systematic way to find the HCF of two numbers. Here's a step-by-step process to find the HCF of 525 and 289:
Divide 525 by 289:525 ÷ 289 1 with a remainder of 236
Divide 289 by 236:289 ÷ 236 1 with a remainder of 53
Divide 236 by 53:236 ÷ 53 4 with a remainder of 24
Divide 53 by 24:53 ÷ 24 2 with a remainder of 5
Divide 24 by 5:24 ÷ 5 4 with a remainder of 4
Divide 5 by 4:5 ÷ 4 1 with a remainder of 1
Divide 4 by 1:4 ÷ 1 4 with a remainder of 0
Since we have reached a remainder of 0, the HCF is the last divisor used, which is 1.
Step-by-Step Verification Using the Euclidean Algorithm
Let's go through the Euclidean algorithm again to ensure we understand the process:
Divide 525 by 289:525 ÷ 289 1 remainder 236
Divide 289 by 236:289 ÷ 236 1 remainder 53
Divide 236 by 53:236 ÷ 53 4 remainder 24
Divide 53 by 24:53 ÷ 24 2 remainder 5
Divide 24 by 5:24 ÷ 5 4 remainder 4
Divide 5 by 4:5 ÷ 4 1 remainder 1
Divide 4 by 1:4 ÷ 1 4 remainder 0
Again, since the remainder is 0, the HCF is the last divisor, which is 1.
Conclusion
The highest common factor (HCF) of 525 and 289 is 1. This can be confirmed through both prime factorization and the Euclidean algorithm. Understanding these methods can help you tackle similar problems in mathematical contexts.