HCF and LCM

Question Sample Titled 'HCF and LCM'

 The H.C.F. and L.C.M. of three expressions are ${9}{a}^{{4}}{b}^{{4}}{c}^{{3}}$ and ${81}{a}^{{7}}{b}^{{7}}{c}^{{6}}$ respectively. If the first expression and the second expression are ${81}{a}^{{{4}}}{b}^{{{7}}}{c}^{{{4}}}$ and ${27}{a}^{{{4}}}{b}^{{{5}}}{c}^{{{6}}}$ respectively, then the third expression is

A
${9}{a}^{{7}}{b}^{{4}}{c}^{{3}}$
B
${27}{a}^{{7}}{b}^{{4}}{c}^{{3}}$
C
${9}{a}^{{3}}{b}^{{4}}{c}^{{3}}$
D
${27}{b}^{{3}}{c}^{{4}}$

 ${81}{a}^{{{4}}}{b}^{{{7}}}{c}^{{{4}}}$ First term ${27}{a}^{{{4}}}{b}^{{{5}}}{c}^{{{6}}}$ Second term $?{a}^{{?}}{b}^{{?}}{c}^{{?}}$ Third term ${9}{a}^{{{4}}}{b}^{{{4}}}{c}^{{{3}}}$ H.C.F. ${81}{a}^{{{7}}}{b}^{{{7}}}{c}^{{{6}}}$ L.C.M. This is equivalent to writing as the followed. ${3}^{{{4}}}{a}^{{{4}}}{b}^{{{7}}}{c}^{{{4}}}$ First term ${3}^{{{3}}}{a}^{{{4}}}{b}^{{{5}}}{c}^{{{6}}}$ Second term ${3}^{{?}}{a}^{{?}}{b}^{{?}}{c}^{{?}}$ Third term ${3}^{{{2}}}{a}^{{{4}}}{b}^{{{4}}}{c}^{{{3}}}$ H.C.F. ${3}^{{{4}}}{a}^{{{7}}}{b}^{{{7}}}{c}^{{{6}}}$ L.C.M. ∴   Third term$={3}^{{{2}}}{a}^{{{7}}}{b}^{{{4}}}{c}^{{{3}}}={9}{a}^{{7}}{b}^{{4}}{c}^{{3}}$

 The highest common factor (HCF) of two polynomials with positive integral indices is the products of all the monomials with smallest integral indices. For examples, the HCF of ${x}^{{2}}{y}^{{3}}$ and ${x}{y}^{{2}}{z}^{{3}}$ is ${x}{y}^{{2}}.$ The least common multiple (LCM) of two polynomials with positive integral indices is the products of all the monomials with largest integral indices. For examples, the LCM of ${x}^{{2}}{y}^{{3}}$ and ${x}{y}^{{2}}{z}^{{3}}$ is ${x}^{{2}}{y}^{{3}}{z}^{{3}}.$

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