Find the greatest 4 digit number which is exactly divisible by 40, 48, and 60

To find the greatest 4-digit number that is exactly divisible by 40, 48, and 60, we first need to determine the least common multiple (LCM) of these three numbers.

1. **Finding the LCM**:

• Prime factorization of each number:
• 40: 2³ × 5
• 48: 2⁴ × 3
• 60: 2² × 3 × 5

2. **Selecting the highest powers of each prime**:

• From 40, we take 2³ and 5.
• From 48, we take 2⁴ and 3.
• From 60, we take 2², 3, and 5.

Thus, the LCM will be:

• 2⁴ (from 48) x 3¹ (from 48 and 60) x 5¹ (from 40 and 60) = 16 x 3 x 5 = 240.

3. **Finding the greatest 4-digit number divisible by 240**:

The greatest 4-digit number is 9999. We can find the number divisible by 240 by calculating:

• 9999 ÷ 240 = 41.6625
• The whole number part is 41. Therefore, we multiply back:
• 41 x 240 = 9840.

Thus, the greatest 4-digit number which is exactly divisible by 40, 48, and 60 is 9840.