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The following steps will be useful to find the greatest common factor of monomials.
Step 1 :
Find the prime factorization of each monomial, including all the variables (the factor '-1' is necessary).
(When the same variable is found in all the monomials, write that variable in each monomial such that it has the same exponent in all the monomials.)
Step 2 :
Multiply those common factors
In the each of the following examples, find the greatest common factor of the monomials.
Example 1 :
7x2yz4 and 21x2y5z3
Solution :
7x2yz4 = 7 ⋅ x2 ⋅ y ⋅ z3 ⋅ z
21x2y5z3 = 3 ⋅ 7 ⋅ x2 ⋅ y ⋅ y4 ⋅ z3
Common factors are shown in blue.
Product of common factors :
= 7 ⋅ x2 ⋅ y ⋅ z3
= 7x2yz3
So, the GCF is 7x2yz3.
Example 2 :
x2y, x3y and x2y2
Solution :
x2y = x2 ⋅ y
x3y = x2 ⋅ x ⋅ y
x2y2 = x2 ⋅ y ⋅ y
Product of common factors :
= x2 ⋅ y
= x2y
So, the GCF is x2y.
Example 3 :
25bc4d3, 35b2c5, and 45c3d
Solution :
25bc4d3 = 5 ⋅ 5 ⋅ b ⋅ c3 ⋅ c ⋅ d3
35b2c5 = 5 ⋅ 7 ⋅ b2 ⋅ c3 ⋅ c2
45c3d = 3 ⋅ 3 ⋅ 5 ⋅ c3 ⋅ d
Product of common factors :
= 5 ⋅ c3
= 5c3
So, the GCF is 5c3.
Example 4 :
35x5y3z4, 49x2yz3 and 14xy2z2
Solution :
35x5y3z4 = 5 ⋅ 7 ⋅ x4 ⋅ x ⋅ y2 ⋅ y ⋅ z2 ⋅ z2
49x2yz3 = 7 ⋅ 7 ⋅ x ⋅ x ⋅ y ⋅ z2 ⋅ z
14xy2z2 = 2 ⋅ 7 ⋅ x ⋅ y ⋅ y ⋅ z2
Product of common factors :
= 7 ⋅ x ⋅ y ⋅ z2
= 7xyz2
So, the GCF is 7xyz2.
Example 5 :
x3y2 and xyz
Solution :
x3y2 = x2 ⋅ x ⋅ y ⋅ y
xyz = x ⋅ y ⋅ z
Product of common factors :
= x ⋅ y
= xy
So, the GCF is xy.
Example 6 :
3x2yz and 4x3y3
Solution :
3x2yz = 3 ⋅ x2 ⋅ y ⋅ z
4x3y3 = 2 ⋅ 2 ⋅ x2 ⋅ x ⋅ y ⋅ y
Product of common factors :
= x2 ⋅ y
= x2y
So, the GCF is x2y.
Example 7 :
p2qr , pq2r and pqr2
Solution :
p2qr = p ⋅ p ⋅ q ⋅ r
pq2r = p ⋅ q ⋅ q ⋅ r
pqr2 = p ⋅ q ⋅ r ⋅ r
Product of common factors :
= p ⋅ q ⋅ r
= pqr
So, the GCF is pqr.
Example 8 :
66a4b2c3, 44a3b4c2 and 24a2b3c4
Solution :
66a4b2c3 = 2 ⋅ 3 ⋅ 11 ⋅ a2 ⋅ a2 ⋅ b2 ⋅ c2 ⋅ c
44a3b4c2 = 2 ⋅ 2 ⋅ 11 ⋅ a2 ⋅ a ⋅ b2 ⋅ b2 ⋅ c2
24a2b3c4 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ a2 ⋅ b2 ⋅ b ⋅ c2 ⋅ c2
Product of common factors :
= 2 ⋅ a2 ⋅ b2 ⋅ c2
= 2a2b2c2
So, the GCF is 2a2b2c2.
Example 9 :
-27p2qr5 and 15p3r3
Solution :
-27p2qr5 = -1 ⋅ 32 ⋅ 3 ⋅ p2 ⋅ q ⋅ r3 ⋅ r2
15p3r3 = 3 ⋅ 5 ⋅ p2 ⋅ p ⋅ r3
Product of common factors :
= 3 ⋅ p2 ⋅ r3
= 3p2r3
So, the GCF is 3p2r3.
Example 10 :
-30x2y2z2 and -50x3y3z3
Solution :
-30x2y2z2 = -1 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ x2 ⋅ y2 ⋅ z2
-50x3y3z3 = -1 ⋅ 2 ⋅ 5 ⋅ 5 ⋅ x2 ⋅ x ⋅ y2 ⋅ y ⋅ z2 ⋅ z
Product of common factors :
= -1 ⋅ 2 ⋅ 5 ⋅ x2 ⋅ y2 ⋅ z2
= -10x2y2z2
So, the GCF is -10x2y2z2.
Example 11 :
The total cost of Baydan and three of her friends to go ice skating can be represented by the expression 4x + 36. The four friends pay an amount x to rent the ice skates and an admission fee, How much is the admission fee for one person ?
Solution :
Total cost paid for ice skating by four friends = 4x + 36
= 4x + 4(9)
Factoring 4, we get
= 4(x + 9)
Example 12 :
Which of the following is the GCF of 16m and 40 mn
a) 8 b) 4m c) 8m d) 4mn
Solution :
16m and 40 mn
Factoring 16 using prime factorization, we get
16 = 24
16m = 24 m
Factoring 40 mn using prime factorization, we get
40 = 2⋅2⋅2⋅5
40 = 23⋅5
40mn = 23⋅5⋅m⋅n
Greatest common factor for these two, we get
23⋅m
= 8m
So, option c is correct.
Example 13 :
What is the GCF of 20x and 25x
Solution :
20x and 25x
20x = 22 ⋅ 5 ⋅ x
25x = 52 ⋅ x
Comparing these two, greatest common factor is 5x.
Example 14 :
What is 10xy - 15y written in factored form ?
a) 5y(2x - 3) b) 5(2x - 3y) c) 10y(x - 3)
Solution :
10xy - 15y
= 2 ⋅ 5 ⋅ x ⋅ y - 3 ⋅ 5 ⋅ y
= 5y(2x - 3)
Example 15 :
Find the greatest common factor for the monomials
48x and 32x
Solution :
48x and 32x
Prime factorization of 48x = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3x
= 24