Greatest Common Factor (GCF) of two numbers is the greatest factor that is common to both of them

To find the greatest common divisor of the given numbers or for algebraic expressions we have to follow the steps.

**Step 1 :**

List the prime factors of each of the given number. For algebraic expression we have to find factors of them.

**Step 2 :**

List the common factors of the given numbers or common factors.

**Step 3 :**

Multiply those common factors.

**Example :**

Find the greatest common divisor of following algebraic terms.

(i) 7 x^{2} y z^{4}, 21 x^{2} y^{5} z^{3}

(ii) x^{2} y, x^{3} y , x^{2} y^{2}

(iii) 25 b c^{4} d^{3} , 35 b^{2} c^{5} , 45 c^{3} d

(iv) 35 x^{5} y^{3} z^{4} , 49 x^{2} y z^{3} , 14 xy^{2} z^{2}

**(i) Answer :**

7 x^{2} y z^{4}, 21 x^{2} y^{5} z^{3}

7 x^{2} y z^{4 }= 7 ⋅ x^{2 }⋅ y ⋅ z ⋅ z^{3}

21 x^{2} y^{5} z^{3 }= 3 ⋅ 7 ⋅ x^{2 }⋅ y ⋅ z ⋅ z^{3}

Common factors are 7, x^{2}, y and z^{3}

Multiplying common factors, we get

= 7x^{2}yz^{3}

So, greatest common divisor of the given algebraic terms is 7x^{2}yz^{3.}

**(ii) Answer :**

x^{2} y, x^{3} y , x^{2} y^{2}

x^{2} y = x^{2 }⋅ y

x^{3} y = x^{2 }⋅ x ⋅ y

x^{2} y^{2 }= x^{2 }⋅ y ⋅ y

Common factors in the above terms are x^{2 }and y.

By multiplying common factors, we get x^{2}y.

So, greatest common divisor of the given algebraic terms is x^{2}y.

**(iii) Answer :**

25 b c^{4} d^{3}, 35 b^{2} c^{5}, 45 c^{3} d

25 b c^{4} d^{3 }= 5^{ }⋅ 5 ⋅ b^{ }⋅ c^{3 }⋅ d^{3}

35 b^{2} c^{5 }= 7^{ }⋅ 5 ⋅ b ⋅ b^{ }⋅ c^{3 }⋅ c^{2}

45 c^{3} d = 3^{2}^{ }⋅ 5 ⋅ c^{3 }⋅ d

Common factors in the above terms are 5 and c^{3}.

By multiplying common factors, we get 5c^{3}.

So, greatest common divisor of the given algebraic terms is 5c^{3}.

**(iv) Answer :**

35 x^{5} y^{3} z^{4} , 49 x^{2} y z^{3} , 14 xy^{2} z^{2}

35 x^{5} y^{3} z^{4} = 7^{ }⋅ 5 ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ y ⋅ y ⋅ y ⋅ z^{2 }⋅ z^{2}

49 x^{2} y z^{3 }= 7^{ }⋅ 7 ⋅ x ⋅ x ⋅ y ⋅ z^{2 }⋅ z

14 xy^{2} z^{2}^{ }= 7^{ }⋅ 2 ⋅ x ⋅ y ⋅ y ⋅ z^{2}

Common factors in the above terms are 7, x, y and z^{2}.

By multiplying common factors, we get 7xyz^{2}.

So, greatest common divisor of the given algebraic terms is 7xyz^{2}.

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