Question

Find the LCM of the following.
(i) $90,150,225$
(ii) $35a^2{c}^{3}b,42a^{3}c{b}^2,30a{c}^2{b}^3$
(iii) $(a-1{)}^5(a+3{)}^2,(a-2{)}^2(a-1{)}^3(a+3{)}^4$
(iv) $x^3+y^3,x^3-y^3,x^4+x^2{y}^2+y^4$

Answer 1

(i) Now, $90=2\times 3\times 3\times 5=2^1\times 3^2\times 5^1$
$150=2\times 3\times 5\times 5=2^1\times 3^1\times 5^1$
$225=3\times 3\times 5\times 5=3^2\times 5^2$
The product $2^1\times 3^2\times 5^2=450$ is the required LCM.

(ii) Now, LCM of 35, 42 and 30 is $5\times 7\times =210$
Hence, the required LCM $=210\times a^3\times c^3\times b^3=210a^3{c}^3{b}^3$

(iii) Now, LCM of $(a-1{)}^5(a+3{)}^2,(a-2{)}^2(a-1{)}^3(a+3{)}^4$ is $(a-1{)}^5(a+3{)}^4(a-2{)}^2$

(iv) Let us first find the factors for each of the given expressions.
$x^3+y^3=(x+y)(x^2-xy+y^2)$
$x^3-y^3=(x-y)(x^2+xy+y^2)$
$x^4+x^2{y}^2+y^4=(x^2+y^2)-x^2{y}^2(x^2+xy+y^2)(x^2-xy+y^2)$
Thus, $(x+y)(x^2-xy+y^2)(x-y)(x^2+xy+y^2)$
$=(x^3+y^3)(x^3-y^3)=x^6-y^6$