Question

using fundamental theorem of arithmetic find the lcm and hcf of 510 and 153​

Answer 1

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

Fundamental Theorem of Arithmetic :-

The fundamental theorem of arithmetic states,

• "Every composite number can be factorized as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur".

To find the HCF and LCM of two numbers, using fundamental theorem of arithmetic.

• 1. For this, we first find the prime factorization of both the numbers.

• 2. HCF is the product of the smallest power of each common prime factor.

• 3. LCM is the product of the greatest power of each common prime factor.

Let's solve the problem now!!

• Prime factorization of 510

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:510\:\:\:\:}}}\\ {\underline{\sf{17}}}& \underline{\sf{\:\:170\: \: \:\:}} \\{\underline{\sf{2}}}& \underline{\sf{\:\:10\: \: \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\: \: \:\:}} \\ {\sf{}}&{\sf{\:\:1\: \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\bf\implies \:510 = 2 \times 3 \times 5 \times 17$

• Prime factorization of 153

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:153\:\:\:\:}}}\\ {\underline{\sf{17}}}& \underline{\sf{\:\:51\: \: \:\:}} \\{\underline{\sf{3}}}& \underline{\sf{\:\:3\: \: \:\:}} \\ {\sf{}}&{\sf{\:\:1\: \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\bf\implies \:153 = {3}^{2} \times 17$

So,

$\bf\implies \:HCF(510, 153) = 17 \times 3 = 51$

and

$\bf\implies \:LCM(510, 153) = 2 \times {3}^{2} \times 5 \times 17 = 1530$