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$${\bf{10}}{{\bf{a}}^{\bf{2}}}{{\bf{b}}^{\bf{2}}}{{\bf{c}}^{\bf{2}}},{\bf{15a}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$$ and $${\bf{20}}{{\bf{b}}^{\bf{3}}}{{\bf{c}}^{\bf{2}}}$$.

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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:

$$2 \times 2 \times 5 \times 3 \times a \times a \times b \times b \times b \times c \times c = 60{a^2}{b^3}{c^2}$$

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.

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