Lowest Common Multiple of Monomials by Factorization

How to find the lowest common multiple of monomials by factorization?

Let us follow the following examples to know how to find the lowest common multiple (L.C.M.) of monomials by factorization.

Solved examples of L.C.M. of monomials by factorization:

1. Find the L.C.M of the monomials 4a²b³ and 12a³b.

Solution:

4a²b³ = 2 × 2 × a × a × b × b × b

12a³b = 2 × 2 × 3 × a × a × a × b

From the resolved factors of the above two monomials, the common factors are indicated by red color.

The common factors among two monomials are 2, 2, a, a, b; other than these common factors, in the first monomial the extra factors are b, b and in the second monomial the extra factors are 3, a.

Therefore, the required L.C.M. = common factors among two monomials × extra common factors among two monomials.

= (2 × 2 × a × a × b)(3 × a × b × b)

= 4a²b × 3ab²

= 12a³b³

Hence, the lowest common multiple of the monomials 4a²b³ and 12a³b = 12a³b³.



2. Find the L.C.M of the monomials 6p²q², 15p³q and 9p²q³r.

Solution:

The L.C.M. of numerical coefficients = The L.C.M. of 6, 15 and 9.

Since, 6 = 2 × 3 = 2¹ × 3¹, 15 = 3 × 5 = 3¹ × 5¹ and 9 = 3 × 3 = 3²

Therefore, the L.C.M. of 6, 15 and 9 is 2¹ × 3² × 5¹ = 2 × 3 × 3 × 5 = 90.

The L.C.M. of literal coefficients = The L.C.M. of p²q², p³q and p²q³r = p³q³r

Since, in p²q², p³q and p²q³r, we get

The highest power of p is p³.

The highest power of q is q³.

The highest power of r is r.

Therefore, the L.C.M. of p²q², p³q and p²q³r = p³q³r.

Thus, the L.C.M. of 6p²q², 15p³q and 9p²q³r

= The L.C.M. of numerical coefficients × The L.C.M. of literal coefficients

= 90 × (p³q³r)

= 90p³q³r.

Note:

According to the well known definition of L.C.M., the expression obtained as L.C.M should be the least expression which should be separately divisible by each and every expression and for this:

(i) the coefficient of the L.C.M. obtained should be equal to the L.C.M. of the coefficient of the given expressions.

(ii) the power of the each variable present in the L.C.M. should be equal to the highest power of that variable present in the given expressions.

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8th Grade Math Practice

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