A largest positive integer which divides two or more integers without any remainder is called Greatest Common Divisor or Greatest Common Factor. |

GCD is the largest positive integer that divides completely given numbers. For example, the GCD of 12 and 30 is 6. There are different methods to find the greatest common divisor. Steps for common division method, prime factorization and listing factor are explained in very simple steps.

**Common Division Method:****1)**Divide the greater number by smaller number and find the remainder.

**2)**Continue this process with the first remainder as divisor and first divisor as dividend.

**3)**Until we get a zero as remainder carry on the process.

**4)**Once we get a zero, last divisor is the highest common factor.

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__Prime factorization__Express the number in terms of its factors.

**1)**Find the prime factors of each of the given numbers.

**2)**Write the common prime factors of the numbers.

**3)**Multiply all the selected prime factors. The product of the common prime factors is the H.C.F.

**Listing factors:****1)**Find the factors of all the numbers.

**2)**Write the factors that are common to all the numbers.

**3)**Write the highest of the common factors. It is the H.C.F.

**Given below are some problems related to greatest common factor. This will help you in having a clear understanding of the topic:**

**Example 1 :**Find the greatest common factors of 45, 12 and 225.

**Solution :**

Factors of 45 = 1, 3, 5, 9, 15, 45

Factors of 12 = 1, 2, 3, 4, 6, 12

Factors of 225 = 1, 3, 5, 9, 12, 25, 45, 75, 225

Therefore the common factors of 45, 12 and 225 = 1 and 3.

So for the given problem highest common factor is 3.

**Example 2 :**Find the H.C.F of 18, 45 and 72?

**Solution :**

Factors of 18 = 1, 2, 3, 6, 9, 18

45 = 1, 3, 5, 9, 15, 45

72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, 72

Common factors of 18, 45 and 72 are 1, 3, 9

Therefore the highest common factor of 18, 45 and 72 is 9.

**Example 3 :**What is the greatest common factor of $\frac{12}{30}$?

**Solution :**First lets find the factor for 12 : 1, 2, 3, 4, 6, 12.

Similarly the factors of 30 are : 1, 2, 3, 5, 6, 10, 15 and 30.

$\Rightarrow$ $\frac{12}{30}$ = $\frac{1, 2, 3, 4, 6, 12}{1, 2, 3, 5, 6, 10, 15, 30}$

From above we can clearly see that the common factors for $\frac{12}{30}$ are 1, 2, 3 and 6.

Therefore the greatest common factor of $\frac{12}{30}$ is 6.