Question

Find the greatest number less than 10,000 which is exactly divisible by 48, 60 and 65.

Hint:

Always verify your final answer by dividing it by the largest of the three given numbers to make sure it is divisible.
Here, \(9360 \div 65 = 144\), which is an integer.
This simple check ensures that your LCM calculation and division steps are correct!

Answer 1

Step 1: Find the LCM of 48 and 60 using their HCF.
The HCF of 48 and 60 is 12, so: \[ \text{LCM}(48,60) = \frac{48 \times 60}{12} = 240 \]
Step 2: Combine this LCM with 65 the same way.
The HCF of 240 and 65 is 5, so: \[ \text{LCM}(240,65) = \frac{240 \times 65}{5} = 3120 \]
Step 3: Divide 10000 by this LCM.
\[ 10000 \div 3120 = 3 \text{ remainder } 640 \]
Step 4: Subtract the remainder to get the greatest multiple below 10000.
\[ 10000 - 640 = 9360 \]
\[ \boxed{9360} \]