Question

Find the LCM and HCF of 2520 and 10530 by prime factorization method.

Hint:

HCF = Product of lowest powers of common primes, and LCM = Product of highest powers of all primes involved.

Answer 1

Step 1: Prime factorization of 2520.
\[ 2520 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 = 2^3 \times 3^2 \times 5 \times 7 \] Step 2: Prime factorization of 10530.
\[ 10530 = 2 \times 3 \times 3 \times 5 \times 5 \times 13 \times 3 \Rightarrow 2 \times 3^2 \times 5 \times 13 \times 3 \] Simplify correctly: \[ 10530 = 2 \times 3^2 \times 5 \times 13 \times 3 \Rightarrow 2 \times 3^2 \times 5 \times 13 \times 3 = 2 \times 3^2 \times 5 \times 13 \] Step 3: Find HCF (common prime factors with least powers).
Common factors are $2^1, 3^2,$ and $5^1$.
\[ \text{HCF} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 = 90 \]
Step 4: Find LCM (all prime factors with highest powers).
\[ \text{LCM} = 2^3 \times 3^2 \times 5 \times 7 \times 13 \] \[ \text{LCM} = 8 \times 9 \times 5 \times 7 \times 13 = 32760 \]
Step 5: Conclusion.
\[ \text{HCF} = 90, \quad \text{LCM} = 32760 \]