Demystifying GCD (Greatest Common Divisor) or HCF (Highest Common Factor)

What is GCD (Greatest Common Divisor) or HCF (Highest Common Factor)?

GCD, which stands for Greatest Common Divisor, and HCF, which stands for Highest Common Factor, refer to the same mathematical concept. It represents the largest number that can exactly divide two or more natural numbers without leaving a remainder.

How is the GCD or HCF calculated for two numbers, n1 and n2?

Calculating the GCD or HCF involves a systematic process:

(a) Find the standard form of the numbers: Express both n1 and n2 in their prime factorization forms.

(b) Write out common prime factors: Identify all the prime factors that are common to the standard forms of both n1 and n2.

(c) Raise each common prime factor to the lesser power: For each prime factor identified in step

(b), use the lowest power at which it appears in the prime factorization of both n1 and n2.

(d) Calculate the product: The GCD or HCF is the product of the results obtained in step (c).

Can you provide an illustration of how to find the GCD or HCF?

Certainly! Let’s find the GCD of 150, 210, and 375:

Step 1: Write down the standard form of the numbers:

• 150 = 5 × 5 × 3 × 2
• 210 = 5 × 2 × 7 × 3
• 375 = 5 × 5 × 5 × 3

Step 2: Identify common prime factors: The common prime factors among the three numbers are 5 and 3.

Step 3: Raise each common prime factor to the lesser power: Since 5 appears as 5 × 5 × 5 in the third number, we use 5 × 5 in the GCD calculation. For 3, it appears with the same power in all three numbers.

Step 4: Calculate the product: The GCD of 150, 210, and 375 is 5 × 3 = 15.

Can you provide more examples for practice?

Certainly! Here are some more examples for finding the GCD or HCF:

(a) 78, 39, 195

• Prime factorization: 78 = 2 × 3 × 13, 39 = 3 × 13, 195 = 3 × 5 × 13
• Common prime factor: 13
• GCD = 13

(b) 440, 140, 390

• Prime factorization: 440 = 2 × 2 × 2 × 5 × 11, 140 = 2 × 2 × 5 × 7, 390 = 2 × 3 × 5 × 13
• Common prime factors: 2 and 5
• GCD = 2 × 5 = 10

(c) 198, 121, 1331

• Prime factorization: 198 = 2 × 3 × 3 × 11, 121 = 11 × 11, 1331 = 11 × 11 × 11
• Common prime factor: 11
• GCD = 11

In summary, the GCD or HCF is the largest number that can exactly divide two or more natural numbers without leaving a remainder. To find it, express the numbers in their prime factorization forms, identify common prime factors, raise them to the lesser power, and calculate their product. This concept is valuable for various mathematical calculations and problem-solving.