Question

what is hcf of324,576,1080 using prime factorzions​

Answer 1

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

$\purple{\rm :\longmapsto\:Prime \: Factorization \: of \: 324}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:324 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:162 \:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:81\:\:}}\\\underline{\sf{3}}&\underline{\sf{\:\:27\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

Hence,

$\purple{\bf :\longmapsto\:Prime \: Factorization \: of \: 324 = {2}^{2} \times {3}^{4} }$

Consider,

$\purple{\bf :\longmapsto\:Prime \: Factorization \: of \: 576 }$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:576 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:288 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:144\:\:}}\\\underline{\sf{2}}&\underline{\sf{\:\:72\:\:}}\\\underline{\sf{2}}&\underline{\sf{\:\:36\:\:}}\\\underline{\sf{2}}&\underline{\sf{\:\:18\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

Hence,

$\purple{\bf :\longmapsto\:Prime \: Factorization \: of \: 576 = {2}^{6} \times {3}^{2} }$

Consider,

$\purple{\bf :\longmapsto\:Prime \: Factorization \: of \: 1080}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:1080 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:540 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:270\:\:}}\\\underline{\sf{3}}&\underline{\sf{\:\:135\:\:}}\\\underline{\sf{3}}&\underline{\sf{\:\:45\:\:}}\\\underline{\sf{3}}&\underline{\sf{\:\:15\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}} \\ \underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

Hence,

$\purple{\bf :\longmapsto\:Prime \: Factorization \: of \:1080 = {2}^{3} \times {3}^{3} \times 5}$

Thus,

We have now,

$\purple{\sf :\longmapsto\:Prime \: Factorization \: of \: 324 = {2}^{2} \times {3}^{4} }$

$\purple{\sf :\longmapsto\:Prime \: Factorization \: of \: 576 = {2}^{6} \times {3}^{2} }$

$\purple{\sf :\longmapsto\:Prime \: Factorization \: of \:1080 = {2}^{3} \times {3}^{3} \times 5}$

$\bf\implies \:HCF (324, 576, 1080) = {2}^{2} \times {3}^{2} = 36$

Additional Information :-

Let a and b are two natural numbers then

• 1. HCF(a, b) × LCM(a, b) = a × b

• 2. HCF always divides a, b and LCM

• 3. LCM is always divisible by HCF