Question

# Find the HCF and LCM of the following Expressions:${\bf{10}}{{\bf{a}}^{\bf{2}}}{{\bf{b}}^{\bf{2}}}{{\bf{c}}^{\bf{2}}},{\bf{15a}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$ and ${\bf{20}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$.

Hint:Factorise all the monomials and then find the highest common factor among all the three factor forms of the monomials to find the HCF then for LCM find the lowest common factor among all the expanded forms of monomials.

Complete step-by-step solution:
The given monomials ${\bf{10}}{{\bf{a}}^{\bf{2}}}{{\bf{b}}^{\bf{2}}}{{\bf{c}}^{\bf{2}}},{\bf{15a}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$ and ${\bf{20}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$can be factored as follows:
\eqalign{ & 10{a^2}{b^2}{c^2} = 2 \times 5 \times a \times a \times b \times b \times c \times c \cr & 15a{b^3}{c^2} = 3 \times 5 \times a \times b \times b \times b \times c \times c \cr & 20{b^3}{c^2} = 2 \times 2 \times 5 \times b \times b \times b \times c \times c \cr}
We know that HCF is the highest common factor, therefore, the HCF of ${\bf{10}}{{\bf{a}}^{\bf{2}}}{{\bf{b}}^{\bf{2}}}{{\bf{c}}^{\bf{2}}},{\bf{15a}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$ and ${\bf{20}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$is:
$5 \times b \times b \times c \times c = 5{b^2}{c^2}$

Also, the LCM is the least common multiple, therefore, the LCM of ${\bf{10}}{{\bf{a}}^{\bf{2}}}{{\bf{b}}^{\bf{2}}}{{\bf{c}}^{\bf{2}}},{\bf{15a}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$ and ${\bf{20}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$is:
$2 \times 2 \times 5 \times 3 \times a \times a \times b \times b \times b \times c \times c = 60{a^2}{b^3}{c^2}$

Hence, the HCF is $5{b^2}{c^2}$ and LCM is $60{a^2}{b^3}{c^2}$.