[tex]{\large{\textsf{\textbf{\underline{\underline{Question \: 1 :}}}}}}[/tex]
[tex]\diamond\:{\boxed{\tt{\purple{{ part \:(i) :}}}}}[/tex]
❍ Prime factorisation :-
‣ Prime factors of 12 -
[tex]\begin{gathered}\begin{gathered}{\begin{array}{ c|c}2&12 \\\hline 2&6 \\ \hline 3&3\\ \hline &1\end{array}} \end{gathered}\end{gathered}[/tex]
‣ Prime factors of 18 -
[tex]\begin{gathered}\begin{gathered}{\begin{array}{ c|c}2&18 \\\hline 3&9\\\hline 3&3 \\ \hline &1\end{array}} \end{gathered}\end{gathered} [/tex]
Therefore,
[tex]\star \: 12 = {2}^{2} \times {3}^{1}[/tex]
[tex]\star \: 18= {2}^{1} \times {3}^{2}[/tex]
✧ LCM of 12 and 18 = [tex]\sf {2}^{2} \times {3}^{2}[/tex]
[tex]\implies 36[/tex]
✧ HCF of 12 and 18 = [tex]\sf {2}^{1} \times {3}^{1}[/tex]
[tex] \implies 6[/tex]
[tex] \diamond\:{\boxed{\tt{\red{{ part \:(ii) :}}}}}[/tex]
❍ Prime factorisation :-
‣ Prime factors of 12 -
[tex]\begin{gathered}\begin{gathered}{\begin{array}{ c|c}2&12 \\\hline 2&6 \\ \hline 3&3\\ \hline &1\end{array}} \end{gathered}\end{gathered}[/tex]
‣ Prime factors of 15 -
[tex]\begin{gathered}\begin{gathered}{\begin{array}{ c|c}3&15 \\\hline 5&5 \\ \hline &1\end{array}} \end{gathered}\end{gathered}[/tex]
‣ Prime factors of 21 -
[tex]\begin{gathered}\begin{gathered}{\begin{array}{ c|c}3&21 \\\hline 7&7 \\ \hline &1\end{array}} \end{gathered}\end{gathered}[/tex]
Therefore,
[tex]\star \: 12 = {2}^{2} \times {3}^{1}[/tex]
[tex] \star \: 15 = {3}^{1} \times {5}^{1}[/tex]
[tex]\star \: 21 = {3}^{1} \times {7}^{1}[/tex]
✧ LCM of 12, 15 and 21 = [tex]\sf {2}^{2} \times {3}^{1} \times {5}^{1} \times {7}^{1}[/tex]
[tex] \implies 420[/tex]
✧ HCF of 12, 15 and 21 = [tex]\sf {3}^{1} [/tex]
[tex] \implies 3[/tex]
[tex] {\large{\textsf{\textbf{\underline{\underline{Question \: 2 :}}}}}}[/tex]
Using,
LCM × HCF = Product of two given numbers
Putting the values,
[tex]\longrightarrow \sf 48 \times 12 = 16 \times y[/tex]
[tex]\longrightarrow \sf \dfrac{ \cancel{48} _{3} \times 12}{ \cancel{16}{ _1} } = y[/tex]
[tex]\longrightarrow \sf y = 3 \times 12[/tex]
[tex] \longrightarrow {\sf{{{\boxed{\green{\bold {\sf y = 36 }}}}}}}[/tex]
Therefore,
The value of y = 36
[tex]{\large{\textsf{\textbf{\underline{\underline{Important \: points :}}}}}}[/tex]
• A natural number "P" > 1 having only two factor one and "P" itself is called prime number.
• A prime number is never negative.
• H.C.F is the product of smallest power of each common prime factor.
• L.C.M is the product of greatest power of each prime factor.
• HCF is always a factor of LCM.
[tex]\underline{\rule{300pts}{3pt}}[/tex]
