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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 $${\bf{20}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$$.

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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 $$60{a^2}{b^3}{c^2}$$.

Additional information:
1. The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.
LCM × HCF = Product of the Numbers
2. HCF of co-prime numbers is 1. Therefore, LCM of given co-prime numbers is equal to the product of the numbers.
LCM of Co-prime Numbers = Product Of The Numbers
3. H.C.F. and L.C.M. of Fractions:
4. LCM of fractions = LCM of numerators / HCF of denominators
    HCF of fractions = HCF of numerators / LCM of denominators
5. HCF of any two or more numbers is never greater than any of the given numbers.
6. LCM of any two or more numbers is never smaller than any of the given numbers.

Note: LCM is the biggest term which should be divisible by all the terms while HCF is the lowest term which can divide all the terms.