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Monomial is an algebra expression that contains one term. It includes numbers, whole numbers, variables, and all are multiplied together. The highest common factor (HCF) or the greatest common factor (GCF) of two monomials are the product of numerical coefficients and literal coefficients.
Steps to Find HCF of Monomials
The following are the simple steps to calculate the greatest common factor of two monomials easily.
- Get the factors of both monomial numerical coefficients.
- From that, find the HCF of the numerical coefficient.
- Coming to the literal coefficients, check the lowest power of each variable.
- And calculate the HCF of literal coefficients.
- Multiply the HCF of numerical coefficients and literal coefficients.
Example Questions on Highest Common Factor of Monomials
Example 1.
Find the H.C.F. of 25x³y² and 5xy³z.
Solution:
The H.C.F. of numerical coefficients = The H.C.F. of 25 and 5.
Since, 25 = 5 × 5 = 5² and 5 = 1 × 5 = 5¹
Therefore, the H.C.F. of 25 and 5 is 5
The H.C.F. of literal coefficients = The H.C.F. of x³y² and xy³z = xy²
Since, in x³y² and xy³z, x and y are common.
The lowest power of x is x.
The lowest power of y is y².
Therefore, the H.C.F. of x³y² and xy³z is xy².
Thus, the H.C.F. of 25x³y² and 5xy³z.
= The H.C.F. of numerical coefficients × The H.C.F. of literal coefficients
= 5 × (xy²)
= 5xy².
Example 2.
Find the G.C.F of 32x²yz² and 72xy²z
Solution:
The H.C.F. of numerical coefficients = The H.C.F. of 32 and 72.
Since, 32 = 2 x 2 x 2 x 2 x 2 = 2⁵ and 72 = 2 x 2 x 2 x 9 = 2³ x 9
Therefore, the H.C.F. of 32 and 72 is 2³ = 8
The H.C.F. of literal coefficients = The H.C.F. of x²yz² and xy²z = xyz
Since, in x²yz² and xy²z, x, y and z are common.
The lowest power of x is x.
The lowest power of y is y.
The lowest power of z is z.
Therefore, the H.C.F. of x³y² and xy³z is xyz.
Thus, the H.C.F. of 32x²yz² and 72xy²z.
= The H.C.F. of numerical coefficients × The H.C.F. of literal coefficients
= 8 × (xyz)
= 8xyz.
Example 3.
Find the highest common factor of 8p²q⁵r³ and 6p⁴q³r².
Solution:
The H.C.F. of numerical coefficients = The H.C.F. of 8 and 6.
Since, 8 = 2 x 2 x 2 = 2³ and 6 = 2 × 3 = 2¹ x 3¹
Therefore, the H.C.F. of 8 and 6 is 2
The H.C.F. of literal coefficients = The H.C.F. of p²q⁵r³ and p⁴q³r² = p²q³r²
Since, in p²q⁵r³ and p⁴q³r², p, q, and r are common.
The lowest power of p is p².
The lowest power of q is q³.
The lowest power of r is r²
Therefore, the H.C.F. of p²q⁵r³ and p⁴q³r² is p²q³r².
Thus, the H.C.F. of 8p²q⁵r³ and 6p⁴q³r².
= The H.C.F. of numerical coefficients × The H.C.F. of literal coefficients
= 2 × (p²q³r²)
= 2p²q³r².
