Three numbers are in the ratio 3 : 4 : 5 and their L.C.M. is 2400. What is the H.C.F. of these three numbers?

Introduction / Context:
This question tests your understanding of the relationship between numbers given in a ratio and their highest common factor (H.C.F.) and least common multiple (L.C.M.). Problems like this appear frequently in aptitude exams to check your grasp of basic number theory concepts involving ratios, H.C.F., and L.C.M.

Given Data / Assumptions:

• The three numbers are in the ratio 3 : 4 : 5.
• Their L.C.M. is 2400.
• We are asked to find the H.C.F. of the three numbers.

Concept / Approach:
If three numbers are in the ratio 3 : 4 : 5, they can be written as 3k, 4k, and 5k for some positive integer k. The L.C.M. of these numbers will be k multiplied by the L.C.M. of 3, 4, and 5. Once we use the given L.C.M. to find k, we can reconstruct the actual numbers and then compute their H.C.F. by inspection, since the H.C.F. will be k itself when the ratio numbers are co-prime to each other.

Step-by-Step Solution:
Step 1: Let the three numbers be 3k, 4k, and 5k.Step 2: Since 3, 4, and 5 are pairwise co-prime, their L.C.M. is 3 * 4 * 5 = 60.Step 3: The L.C.M. of 3k, 4k, and 5k is therefore 60k.Step 4: We are told that the L.C.M. is 2400, so 60k = 2400.Step 5: Solve for k: k = 2400 / 60 = 40.Step 6: So the actual numbers are 3 * 40 = 120, 4 * 40 = 160, and 5 * 40 = 200.Step 7: The H.C.F. of 120, 160, and 200 is clearly 40, since all three are multiples of 40 and no larger common factor exists.

Verification / Alternative check:
Check with prime factors: 120 = 2^3 * 3 * 5, 160 = 2^5 * 5, 200 = 2^3 * 5^2.The common part is 2^3 * 5 = 8 * 5 = 40, which matches our H.C.F. This confirms that 40 is the correct highest common factor.

Why Other Options Are Wrong:
Option b (80) is too large because 80 does not divide 120 exactly. Option c (120) also fails because 120 does not divide 160 or 200 without leaving remainders. Option d (200) is larger than one of the numbers (160), so it cannot be a common factor. Only 40 divides all three numbers exactly.

Common Pitfalls:
A common mistake is to confuse ratio numbers with the actual numbers, taking 3, 4, and 5 themselves as the numbers. Another error is to miscalculate the L.C.M. of 3, 4, and 5 or to forget that the L.C.M. of the scaled numbers is the L.C.M. of the ratio multiplied by k. Some students also try to guess the H.C.F. directly from the L.C.M. without using the ratio properly, which leads to wrong answers.

Final Answer:
The H.C.F. of the three numbers is 40.