Solution:
First,
Find the prime factors of the given integers:
![\[84,\text{ }90\text{ }and\text{ }120\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8ca4dd904f0c9da751da2d679a0d3b2f_l3.png "Rendered by QuickLaTeX.com")
For,
![\[\begin{array}{*{35}{l}} <!-- /wp:paragraph --> <!-- wp:paragraph --> ~84\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }7 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> 90\text{ }=\text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }5 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> 120\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }5 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-cd5a5d55acf9dd218add8c3d95b6fcb5_l3.png "Rendered by QuickLaTeX.com")
Now,
![\[\begin{array}{*{35}{l}} <!-- /wp:paragraph --> <!-- wp:paragraph --> L.C.M\text{ }of\text{ }84,\text{ }90\text{ }and\text{ }120\text{ }=\text{ }{{2}^{3}}~\times \text{ }{{3}^{2}}~\times \text{ }5\text{ }\times \text{ }7 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> \therefore L.C.M\text{ }\left( 84,\text{ }90,\text{ }120 \right)\text{ }=\text{ }2520 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-932c6d2fdc49f74da76bce283a741003_l3.png "Rendered by QuickLaTeX.com")
And,
![\[\begin{array}{*{35}{l}} <!-- /wp:paragraph --> <!-- wp:paragraph --> ~ \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> ~H.C.F\text{ }\left( 84,\text{ }90\text{ }and\text{ }120 \right)\text{ }=\text{ }6 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-19b4fa74c49a3990c21e9ce5079225ac_l3.png "Rendered by QuickLaTeX.com")
Solution:
First,
find the prime factors of the given integers:
![\[24,\text{ }15\text{ }and\text{ }36\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-25d4af426ff65e336a04a6427913009c_l3.png "Rendered by QuickLaTeX.com")
For,
![\[\begin{array}{*{35}{l}} <!-- /wp:paragraph --> <!-- wp:paragraph --> 24\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }x\text{ }2\text{ }x~3 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> 15\text{ }=\text{ }3\text{ }\times \text{ }5 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> 36\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3 \\ <!-- /wp:paragraph --> <!-- wp:paragraph --> \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b1dbdaa2de8e60f76e0ab43c2243991d_l3.png "Rendered by QuickLaTeX.com")
Now,
![\[LCM\text{ }of\text{ }24,\text{ }15\text{ }and\text{ }36\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }5\text{ }=\text{ }{{2}^{3}}~x\text{ }{{3}^{2}}~x\text{ }5\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-01c07c565707dae7d04a188a28478539_l3.png "Rendered by QuickLaTeX.com")
∴
![\[LCM\text{ }\left( 24,\text{ }15,\text{ }36 \right)\text{ }=\text{ }360\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-20c7665e55ddd297f13f7c8b8c2aa239_l3.png "Rendered by QuickLaTeX.com")
And,
![\[HCF\text{ }\left( 24,\text{ }15\text{ }and\text{ }36 \right)\text{ }=\text{ }3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fa607da8c0affaa892fd1fb2b3b05ce1_l3.png "Rendered by QuickLaTeX.com")
2. Find the LCM and HCF of the following integers by applying the prime factorization method:(v)
![\[\mathbf{84},\text{ }\mathbf{90}\text{ }\mathbf{and}\text{ }\mathbf{120}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ccfecca6ebaa7e44c50bda4ca8c75939_l3.png "Rendered by QuickLaTeX.com")
![\[\mathbf{24},\text{ }\mathbf{15}\text{ }\mathbf{and}\text{ }\mathbf{36}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-94f859fd8a71046e251dab20e2d08dcb_l3.png "Rendered by QuickLaTeX.com")
