LCM of 45,60,and 75Solve the problem please

1.	LCM of 45,60,and 75Solve the problem please
Answer» $\large\underline{\bf{Solution-}}$ $\red{\bf :\longmapsto\:Prime \: factorization \: of \: 45}$ $\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c\|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:45\:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}} \\{\sf{}}&\underline{\sf{\:\:1\:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$ $\red{\bf :\longmapsto\:Prime \: factorization \: of \: 45 = {3}^{2} \times 5}$ $\blue{\bf :\longmapsto\:Prime \: factorization \: of \: 60}$ $\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c\|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:60\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:30 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}} \\{\sf{}}&\underline{\sf{\:\:1\:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$ $\blue{\bf :\longmapsto\:Prime \: factorization \: of \: 60 = {2}^{2} \times 3 \times 5}$ $\green{\bf :\longmapsto\:Prime \: factorization \: of \: 75}$ $\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c\|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:75\:\:}}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3 \:\:}} \\ {\sf{}}&\underline{\sf{\:\:1\:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$ $\green{\bf :\longmapsto\:Prime \: factorization \: of \: 75 = {5}^{2} \times 3}$ Hence, $\red{ \bf \: <klux>LCM</klux>(45,60,75) = {5}^{2} \times {3}^{2} \times {2}^{2} = 900}$ Additional Information :- Let us consider two natural NUMBERS a and b, then 1. HCF(a, b) × LCM(a, b) = a × b 2. HCF(a, b) < = (a, b) 3. LCM(a, b) > = (a, b) 4. HCF is a factor of LCM.

Answer»

$\large\underline{\bf{Solution-}}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 45}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:45\:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}} \\{\sf{}}&\underline{\sf{\:\:1\:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 45 = {3}^{2} \times 5}$

$\blue{\bf :\longmapsto\:Prime \: factorization \: of \: 60}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:60\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:30 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}} \\{\sf{}}&\underline{\sf{\:\:1\:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\blue{\bf :\longmapsto\:Prime \: factorization \: of \: 60 = {2}^{2} \times 3 \times 5}$

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 75}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:75\:\:}}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3 \:\:}} \\ {\sf{}}&\underline{\sf{\:\:1\:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 75 = {5}^{2} \times 3}$

Hence,

$\red{ \bf \: <klux>LCM</klux>(45,60,75) = {5}^{2} \times {3}^{2} \times {2}^{2} = 900}$

Additional Information :-

Let us consider two natural NUMBERS a and b, then

1. HCF(a, b) × LCM(a, b) = a × b

2. HCF(a, b) < = (a, b)

3. LCM(a, b) > = (a, b)

4. HCF is a factor of LCM.