Question

Find the H.C.F. and L.C.M. of 1530 and 2040.

Hint:

You can quickly check your answer using: \[ \text{H.C.F.} \times \text{L.C.M.} = 1530 \times 2040 \] Here, \(510 \times 6120 = 3,121,200\), and \(1530 \times 2040 = 3,121,200\).
Since both products match, the calculation is verified to be accurate!

Answer 1

Correct Answer: 6120

Step 1: Prime factorise 1530.
\(1530 = 2 \times 765 = 2 \times 3 \times 255 = 2 \times 3 \times 3 \times 85 = 2 \times 3^2 \times 5 \times 17\).
Step 2: Prime factorise 2040.
\(2040 = 2 \times 1020 = 2^2 \times 510 = 2^2 \times 2 \times 255 = 2^3 \times 3 \times 5 \times 17\).
Step 3: Find the H.C.F.
H.C.F. = product of the lowest powers of common prime factors. Common primes: 2, 3, 5, 17. \(\text{HCF} = 2^1 \times 3^1 \times 5^1 \times 17^1 = 2 \times 3 \times 5 \times 17 = 510\).
Step 4: Find the L.C.M.
L.C.M. = product of the highest powers of all prime factors. \(\text{LCM} = 2^3 \times 3^2 \times 5^1 \times 17^1 = 8 \times 9 \times 5 \times 17 = 6120\).
Step 5: Verify using the relation HCF x LCM = Product of numbers.
\(510 \times 6120 = 3{,}121{,}200\). Also \(1530 \times 2040 = 3{,}121{,}200\). Verified!
Step 6: State the final answers.
\[ \boxed{\text{HCF} = 510, \quad \text{LCM} = 6120} \]