The L.C.M. of two numbers is 2376 and their H.C.F. is 33. If one number is 297, find the other number.

Introduction / Context:
The fundamental identity for two positive integers a and b is a*b = gcd(a, b) * lcm(a, b). Knowing the LCM, HCF, and one number allows us to compute the other directly by rearrangement.

Given Data / Assumptions:

• LCM = 2376.
• HCF = 33.
• One number a = 297.
• Find the other number b.

Concept / Approach:
Use b = (HCF * LCM) / a. Ensure a divides the product exactly, since the result must be an integer. Perform simplification before multiplication if convenient.

Step-by-Step Solution:

Verification / Alternative check:
Check gcd(297, 264) = 33 and lcm(297, 264) = (297*264)/33 = 2376. Both match the given data, confirming correctness.

Why Other Options Are Wrong:

• 216, 642, 792, 198: These do not satisfy both the specified HCF and LCM simultaneously with 297 under the product identity.

Common Pitfalls:

• Miscalculating the quotient 2376/297.
• Forgetting to divide by the known number after multiplying HCF and LCM.

Final Answer:
264