To solve the problem of dividing the LCM of 240, 320, and 460 by their HCF, we will follow these steps:
- Find the HCF (Highest Common Factor) of 240, 320, and 460.
- Find the LCM (Lowest Common Multiple) of 240, 320, and 460.
- Divide the LCM by the HCF.
Step 1: Find the HCF
To find the HCF, we can use the prime factorization method.
-
240:
- \( 240 = 2^4 \times 3^1 \times 5^1 \)
-
320:
- \( 320 = 2^6 \times 5^1 \)
-
460:
- \( 460 = 2^2 \times 5^1 \times 23^1 \)
Now, we take the lowest power of each prime factor present in all three numbers:
- For \( 2 \), the lowest power is \( 2^2 \).
- For \( 3 \), it appears only in 240, not in the other numbers (so not considered).
- For \( 5 \), the lowest power is \( 5^1 \).
- For \( 23 \), it appears only in 460 (so not considered).
Thus, the HCF is: \[ HCF = 2^2 \times 5^1 = 4 \times 5 = 20 \]
Step 2: Find the LCM
To calculate the LCM, we take the highest power of each prime factor that appears in any of the numbers:
- For \( 2 \), the highest power is \( 2^6 \).
- For \( 3 \), the highest power is \( 3^1 \) (from 240).
- For \( 5 \), the highest power is \( 5^1 \) (common in all).
- For \( 23 \), the highest power is \( 23^1 \) (from 460).
Thus, the LCM is: \[ LCM = 2^6 \times 3^1 \times 5^1 \times 23^1 \]
Calculating that:
- \( 2^6 = 64 \)
- \( 3^1 = 3 \)
- \( 5^1 = 5 \)
- \( 23^1 = 23 \)
Now we calculate the LCM: \[ LCM = 64 \times 3 \times 5 \times 23 \]
Calculating it step by step:
- \( 64 \times 3 = 192 \)
- \( 192 \times 5 = 960 \)
- \( 960 \times 23 = 22080 \)
So, \( LCM = 22080 \).
Step 3: Divide the LCM by the HCF
Now, we will divide the LCM by the HCF: \[ \frac{LCM}{HCF} = \frac{22080}{20} \]
Calculating that: \[ \frac{22080}{20} = 1104 \]
Final Answer:
Thus, the result of dividing the LCM of 240, 320, and 460 by their HCF is \( \boxed{1104} \).