Highest Common Factor (HCF) by Division Method #HCFBYDIVISIONMETHOD #GREATESTCOMMONDIVISOR @ZOI-F8Z

Understanding Highest Common Factor (HCF) The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more integers is the largest positive integer that divides each of them without leaving a remainder. The division method is an efficient way to compute the HCF, based on the Euclidean algorithm. This algorithm uses repeated division to simplify the problem until the HCF is found. Steps of the Division Method (Euclidean Algorithm) for Two Numbers To find the HCF of two numbers, say a and bwhere a greater than b: 1. Divide the larger number a by the smaller number b, and find the remainder r. (That is, a = b times q + r, where q is the quotient.) 2. Replace a with b, and b with the remainder r. Repeat the process until the remainder r becomes 0. 3. The last non-zero remainder (or the divisor when remainder is 0) is the HCF. This works because the HCF of a and b is the same as the HCF of b and r, and the process reduces the numbers quickly. Example: Finding HCF of 48 and 18 Let's apply the method step by step: Start with a = 48, b = 18. Divide 48 by 18: 48 div 18 = 2 (quotient), remainder 48 - (18 times 2) = 48 - 36 = 12. Now, replace: a = 18, b = 12. Divide 18 by 12: 18 div 12 = 1 (quotient), remainder 18 - (12 times 1) = 6. Replace: a = 12, b = 6. Divide 12 by 6: 12 div 6 = 2 (quotient), remainder 12 - (6 times 2) = 0. Remainder is 0, so HCF is the last divisor, which is 6. Verification: Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 18 are 1, 2, 3, 6, 9, 18. Common factors: 1, 2, 3, 6. Highest is 6. Extending to More Than Two Numbers To find HCF of three or more numbers (e.g., a, b, c: First, find HCF of the first two (say HCF of a and b = d. Then, find HCF of d and c. Continue this way for additional numbers. Example: HCF of 48, 18, and 30. HCF(48, 18) = 6 (from above). Now HCF(6, 30): Divide 30 by 6 → remainder 0, so HCF = 6. This method is efficient and can be implemented easily in code or by hand for larger numbers