Lowest Common Factor of Monomials
How
to find the lowest common multiple of monomials?
To find the lowest common multiple (L.C.M.) of two or more
monomials is the product of the L.C.M. of their numerical coefficients and the
L.C.M. of their literal coefficients.
Note: The L.C.M. of literal
coefficients is each literal contained in the expression with the highest
power.
Solved
examples to find lowest common multiple of monomials:
1. Find the L.C.M. of 24x
^{3}y
^{2}z and 30x
^{2}y
^{3}z
^{4}.
Solution:
The L.C.M. of numerical coefficients = The L.C.M. of 24 and 30.
Since, 24 = 2 × 2 × 2 × 3 = 2
^{3} × 3
^{1} and 30 = 2 × 3 × 5 = 2
^{1} × 3
^{1} × 5
^{1}
Therefore, the L.C.M. of 24 and 30 is 2
^{3} × 3
^{1} × 5
^{1} = 2 × 2 × 2 × 3 × 5 = 120
The L.C.M. of literal coefficients = The L.C.M. of x
^{3}y
^{2}z and x
^{2}y
^{3}z
^{4} = x
^{3}y
^{3}z
^{4}
Since, in x
^{3}y
^{2}z and x
^{2}y
^{3}z
^{4},
The highest power of x is x
^{3}.
The highest power of y is y
^{3}.
The highest power of z is z
^{4}.
Therefore, the L.C.M. of x
^{3}y
^{2}z and x
^{2}y
^{3}z
^{4} = x
^{3}y
^{3}z
^{4}.
Thus, the L.C.M. of 24x
^{3}y
^{2}z and 30x
^{2}y
^{3}z
^{4}
= The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients
= 120 × (x
^{3}y
^{3}z
^{4})
= 120x
^{3}y
^{3}z
^{4}.
2. Find the L.C.M. of 18x
^{2}y
^{2}z
^{3} and 16xy
^{2}z
^{2}.
Solution:
The L.C.M. of numerical coefficients = The L.C.M. of 18 and 16.
Since, 18 = 2 × 3 × 3 = 2
^{1} × 3
^{2} and 16 = 2 × 2 × 2 × 2 = 2
^{4}
Therefore, the L.C.M. of 18 and 16 is 2
^{4} × 3
^{2} = 2 × 2 × 2 × 2 × 3 × 3 = 144
The L.C.M. of literal coefficients = The L.C.M. of x
^{2}y
^{2}z
^{3} and xy
^{2}z
^{2} = x
^{2}y
^{2}z
^{3}
Since, in x
^{2}y
^{2}z
^{3} and xy
^{2}z
^{2},
The highest power of x is x
^{2}.
The highest power of y is y
^{2}.
The highest power of z is z
^{3}.
Therefore, the L.C.M. of x
^{2}y
^{2}z
^{3} and xy
^{2}z
^{2} = x
^{2}y
^{2}z
^{3}.
Thus, the L.C.M. of 18x
^{2}y
^{2}z
^{3} and 16xy
^{2}z
^{2}
= The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients
= 144 × (x
^{2}y
^{2}z
^{3})
= 144x
^{2}y
^{2}z
^{3}.
8th Grade Math Practice
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