Understanding the Problem
To find the H.C.F (Highest Common Factor) of numbers in the ratio of 4:5:6, with a given L.C.M (Lowest Common Multiple) of 2400, we can break it down into steps.
Step 1: Expressing the Numbers
- Let the numbers be 4x, 5x, and 6x, where x is a common factor.
Step 2: Finding the L.C.M
- The L.C.M of these numbers can be calculated as follows:
- L.C.M(4x, 5x, 6x) = L.C.M(4, 5, 6) * x
- The L.C.M of 4, 5, and 6 is 60.
- Therefore, we have:
- L.C.M(4x, 5x, 6x) = 60x
Step 3: Setting Up the Equation
- Given that the L.C.M is 2400, we set up the equation:
- 60x = 2400
Step 4: Solving for x
- Dividing both sides by 60 gives:
- x = 2400 / 60
- x = 40
Step 5: Finding the Actual Numbers
- Now, we can find the actual numbers:
- First number = 4x = 4 * 40 = 160
- Second number = 5x = 5 * 40 = 200
- Third number = 6x = 6 * 40 = 240
Step 6: Finding the H.C.F
- Finally, we calculate the H.C.F of 160, 200, and 240.
- The H.C.F can be determined using prime factorization:
- 160 = 2^5 * 5^1
- 200 = 2^3 * 5^2
- 240 = 2^4 * 3^1 * 5^1
- Taking the lowest powers of common prime factors:
- H.C.F = 2^3 * 5^1 = 8 * 5 = 40
Conclusion
- The H.C.F of the numbers in the ratio 4:5:6 with an L.C.M of 2400 is 40.