Let's find L.C.M. of 4a²b⁴c, 12a³bc⁵ and 18a²b³c²

Let us understand what L.C.M, Least Common Multiple is.

Least common multiple of two or more integers, is the smallest positive integer that is divisible by these two or more integers.

In order to find L.C.M, we need to find factors of each terms.

Factorization of 4a²b⁴c = 2 × 2 × a × a × b × b × b × b × c

Or, Factorization of 4a²b⁴c = 2² × a² × b⁴ × c

Factorization of 12a³bc⁵ = 2 × 2 × 3 × a × a × a × b × c × c × c × c × c

Or, Factorization of 12a³bc⁵ = 2² × 3 × a³ × b × c⁵

Factorization of 18a²b³c² = 2 × 3 × 3 × a × a × b × b × b × c × c

Or, Factorization of 18a²b³c² = 2 × 3² × a² × b³ × c²

Now, we need to find the factor of highest power.

Factors of 4a²b⁴c = 2², a², b⁴, c

Factors of 12a³bc⁵ = 2², 3, a³, b, c⁵

Factors of 18a²b³c² = 2, 3², a², b³, c²

The factors are 2², 3², a³, b⁴ and c⁵.

By multiplying these factors, we get

2² × 3² × a³ × b⁴ × c⁵ = 4 × 9 a³b⁴c⁵

⇒ 2² × 3² × a³ × b⁴ × c⁵ = 36 a³b⁴c⁵

Thus, lcm of 4a²b⁴c, 12a³bc⁵ and 18a²b³c² is 36 a³b⁴c⁵.