Question

Find HCF of 196,216,120 by prime factorization​

Answer 1

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

Consider,

$\red{\rm :\longmapsto\:Prime \: factorization \: of \: 120}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:120 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:60 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:30\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

Thus,

$\red{\rm :\longmapsto\:Prime \: factorization \: of \: 120 = {2}^{3} \times 3 \times 5 }$

Consider,

$\blue{\rm :\longmapsto\:Prime \: factorization \: of \: 196}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:196 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:98 \:\:}} \\\underline{\sf{7}}&\underline{\sf{\:\:49\:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:7\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

Thus,

$\blue{\rm :\longmapsto\:Prime \: factorization \: of \: 196 = {2}^{2} \times {7}^{2} }$

Consider,

$\green{\rm :\longmapsto\:Prime \: factorization \: of \: 216}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:216 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:108 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:54\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27 \:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}$

Thus,

$\green{\rm :\longmapsto\:Prime \: factorization \: of \: 216 = {2}^{3} \times {3}^{3} }$

Now, we have

$\red{\rm :\longmapsto\:Prime \: factorization \: of \: 120 = {2}^{3} \times 3 \times 5 }$

$\blue{\rm :\longmapsto\:Prime \: factorization \: of \: 196 = {2}^{2} \times {7}^{2} }$

$\green{\rm :\longmapsto\:Prime \: factorization \: of \: 216 = {2}^{3} \times {3}^{3} }$

Hence,

$\purple{\bf :\longmapsto\:HCF(120,196,216) = {2}^{2} = 4}$

Additional Information

If a and b are two positive integers having HCF 'H' and having LCM 'L', then

1. H × L = a × b

2. H always divides L

3. H always divides a

4. H always divides b

5. H is always less than or equal to either a or b.

6. L is always greater than or equals to either a or b.