Find the greatest 4-digit number which is exactly divisible by the numbers 4, 15 and 20.

Answer

To find the greatest 4-digit number that is exactly divisible by the numbers 4, 15, and 20, we can follow these steps:

### Step 1: Identify the greatest 4-digit number
The greatest 4-digit number is 9999.

### Step 2: Find the least common multiple (LCM) of the numbers 4, 15, and 20
To find a number that is divisible by 4, 15, and 20, we first need to calculate the LCM of these numbers.

- **Prime factorization**:
- 4 = 2²
- 15 = 3 × 5
- 20 = 2² × 5

- **Identify the highest power of each prime factor**:
- The highest power of 2 is 2² (from 4 and 20).
- The highest power of 3 is 3¹ (from 15).
- The highest power of 5 is 5¹ (from 15 and 20).

- **Calculate the LCM**:
\[ \text{LCM} = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60 \]

### Step 3: Find the greatest 4-digit number divisible by the LCM
Now we need to find the largest 4-digit number that is divisible by 60.

- We can do this by dividing 9999 by 60 and taking the floor of the result:
\[ 9999 ÷ 60 = 166.65 \]
- The largest integer less than or equal to 166.65 is 166.

### Step 4: Multiply the integer by the LCM
Now, multiply 166 by 60 to find the greatest 4-digit number that is divisible by 60:
\[ 166 × 60 = 9960 \]

### Conclusion
The greatest 4-digit number which is exactly divisible by 4, 15, and 20 is **9960**.

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