The H.C.F. of two numbers is 8.\nWhich of the following values can never be the L.C.M. of these two numbers?

Introduction:
This question examines your understanding of the basic relationship between the highest common factor (H.C.F.) and the least common multiple (L.C.M.) of two integers. It focuses on necessary divisibility conditions that any valid L.C.M. must satisfy when the H.C.F. is given.

Given Data / Assumptions:

• H.C.F. of two numbers = 8.
• We have four candidate values for the L.C.M.: 32, 48, 60, 152, and an extra option 88.
• We must determine which value cannot possibly be the L.C.M.

Concept / Approach:
Key facts:

• The H.C.F. of two integers always divides their L.C.M.
• Therefore, if H.C.F. = 8, then any possible L.C.M. must be a multiple of 8.
• If a number is not divisible by 8, it cannot be the L.C.M. of two numbers whose H.C.F. is 8.

Step-by-Step Solution:
Step 1: Check each option for divisibility by 8. Step 2: 32 ÷ 8 = 4 (integer) → possible L.C.M. Step 3: 48 ÷ 8 = 6 (integer) → possible L.C.M. Step 4: 60 ÷ 8 = 7.5 (not an integer) → cannot be L.C.M. Step 5: 152 ÷ 8 = 19 (integer) → possible L.C.M. Step 6: 88 ÷ 8 = 11 (integer) → possible L.C.M. Step 7: Hence, 60 is the only option that is not divisible by 8.

Verification / Alternative check:
For an L.C.M. candidate L and H.C.F. H, the product of the two numbers a and b is H * L. If H does not divide L, there is no way to get integer values for a and b with H as their H.C.F. Therefore, any candidate not divisible by 8 is automatically invalid as an L.C.M.

Why Other Options Are Wrong:
32: It is a multiple of 8, so it can be an L.C.M. for suitable numbers such as 8 and 32. 48: Also a multiple of 8, so valid for some pair of numbers with H.C.F. 8. 152: Divisible by 8, hence it can be an L.C.M. for appropriate numbers. 88: Another multiple of 8 and thus can serve as an L.C.M.

Common Pitfalls:
Students often forget the simple rule that H.C.F. must divide L.C.M. and may attempt unnecessarily complex calculations. Others confuse H.C.F. and L.C.M. relationships or only check whether the L.C.M. is larger than the H.C.F., which is not sufficient.

Final Answer:
The value that can never be the L.C.M. of two numbers with H.C.F. 8 is 60.