Two numbers have H.C.F. (greatest common divisor) 44 and L.C.M. (least common multiple) 264. The first number is 88 (since 88 ÷ 2 = 44). Find the other number.

Introduction / Context:
The product of two positive integers equals the product of their GCD and LCM. Given the GCD (H.C.F.), LCM, and one number, we can find the other number directly from this relationship. Here, the first number is clarified as 88 via the quotient condition “first number divided by 2 equals 44.”

Given Data / Assumptions:

• HCF = 44.
• LCM = 264.
• First number a = 88 (since 88 ÷ 2 = 44).
• Second number b = ?

Concept / Approach:
Use the identity a*b = HCF * LCM. Rearranging gives b = (HCF * LCM) / a. Substitute the known values and compute b. This method is standard and robust for such problems.

Step-by-Step Solution:

Verification / Alternative check:
Check gcd(88, 132) = 44 and lcm(88, 132) = (88*132)/44 = 264. Both match the given, confirming correctness.

Why Other Options Are Wrong:

• 264, 66, 33, 176: Do not satisfy both gcd and lcm simultaneously with 88 under the product identity.

Common Pitfalls:

• Mistakes in multiplying HCF and LCM or dividing by the first number.
• Forgetting that a and b must be integers compatible with the given gcd and lcm.

Final Answer:
132