To find the largest common divisor (GCD) of the numbers 1026, 2268, and 2430, we can use the method of prime factorization. The GCD is the product of the common prime factors, each raised to the lowest power it appears in any of the factorizations.

Understanding Largest Common Divisor (GCD)

The largest common divisor, also known as the greatest common divisor (GCD) or highest common factor (HCF), of two or more non-zero integers is the largest positive integer that divides each of the integers without leaving a remainder.

Prime Factorization of the Numbers

Let's find the prime factors for each number:

Prime Factorization of 1026

We divide 1026 by the smallest prime numbers until we reach 1:

• $1026 \div 2 = 513$
• $513 \div 3 = 171$
• $171 \div 3 = 57$
• $57 \div 3 = 19$
• $19 \div 19 = 1$

So, the prime factorization of 1026 is $2^1 \times 3^3 \times 19^1$.

Prime Factorization of 2268

We divide 2268 by the smallest prime numbers until we reach 1:

• $2268 \div 2 = 1134$
• $1134 \div 2 = 567$
• $567 \div 3 = 189$
• $189 \div 3 = 63$
• $63 \div 3 = 21$
• $21 \div 3 = 7$
• $7 \div 7 = 1$

So, the prime factorization of 2268 is $2^2 \times 3^4 \times 7^1$.

Prime Factorization of 2430

We divide 2430 by the smallest prime numbers until we reach 1:

• $2430 \div 2 = 1215$
• $1215 \div 3 = 405$
• $405 \div 3 = 135$
• $135 \div 3 = 45$
• $45 \div 3 = 15$
• $15 \div 3 = 5$
• $5 \div 5 = 1$

So, the prime factorization of 2430 is $2^1 \times 3^5 \times 5^1$.

Finding Common Prime Factors and Lowest Powers

Now, let's look at the prime factorizations of all three numbers:

• $1026 = 2^1 \times 3^3 \times 19^1$
• $2268 = 2^2 \times 3^4 \times 7^1$
• $2430 = 2^1 \times 3^5 \times 5^1$

The common prime factors are 2 and 3.

Let's find the lowest power for each common prime factor across the three numbers:

• For prime factor 2: The powers are $2^1$ (in 1026), $2^2$ (in 2268), and $2^1$ (in 2430). The lowest power is $2^1$.
• For prime factor 3: The powers are $3^3$ (in 1026), $3^4$ (in 2268), and $3^5$ (in 2430). The lowest power is $3^3$.

Calculating the GCD

The GCD is the product of these common prime factors raised to their lowest powers:

GCD$(1026, 2268, 2430) = 2^1 \times 3^3 = 2 \times (3 \times 3 \times 3) = 2 \times 27 = 54$.

Thus, the largest common divisor of 1026, 2268, and 2430 is 54.

Revision Table: GCD Calculation Summary

Additional Information: Other Methods for GCD

Besides prime factorization, another common method to find the GCD is the Euclidean Algorithm. This method is generally more efficient for larger numbers.

Euclidean Algorithm

The Euclidean Algorithm for two numbers works by repeatedly applying the division lemma:

For two positive integers $a$ and $b$ with $a > b$, GCD$(a, b)$ = GCD$(b, r)$, where $r$ is the remainder when $a$ is divided by $b$. The process continues until the remainder is 0. The last non-zero remainder is the GCD.

To find the GCD of three numbers (a, b, c), you can first find GCD(a, b) and then find the GCD of the result and c: GCD(a, b, c) = GCD(GCD(a, b), c).

Least Common Multiple (LCM)

Related to GCD is the Least Common Multiple (LCM). The LCM of two or more numbers is the smallest positive integer that is a multiple of all the numbers. For two numbers $a$ and $b$, LCM$(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}$. For multiple numbers, you can use prime factorization: the LCM is the product of all unique prime factors raised to their highest power.