What is the greatest 3-digit number divisible by 4, 5 and 6?

Options

A. a) 990
B. b) 930
C. c) 960
D. d) 900

Answer

To find the greatest 3-digit number that is divisible by 4, 5, and 6, we can follow these steps:

### Step 1: Find the LCM of 4, 5, and 6
To determine if a number is divisible by multiple numbers, we can find the Least Common Multiple (LCM) of those numbers.

- The prime factorization of each number is:
- 4 = 2²
- 5 = 5¹
- 6 = 2¹ × 3¹

- To find the LCM, we take the highest power of each prime factor:
- For 2, the highest power is 2² (from 4)
- For 3, the highest power is 3¹ (from 6)
- For 5, the highest power is 5¹ (from 5)

Thus, the LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60.

### Step 2: Identify the greatest 3-digit number
The greatest 3-digit number is 999.

### Step 3: Divide 999 by the LCM (60)
Now, we need to find the largest multiple of 60 that is less than or equal to 999.

- Divide 999 by 60:
- 999 ÷ 60 = 16.65 (approximately)

### Step 4: Find the greatest integer multiple
The greatest integer less than or equal to 16.65 is 16.

### Step 5: Calculate the greatest multiple of 60
Now, we multiply 60 by 16 to find the greatest 3-digit number that is divisible by 60:
- 60 × 16 = 960.

### Step 6: Verify divisibility
Finally, we can check if 960 is divisible by 4, 5, and 6:
- 960 ÷ 4 = 240 (divisible)
- 960 ÷ 5 = 192 (divisible)
- 960 ÷ 6 = 160 (divisible)

Since 960 is divisible by all three numbers, we conclude that:

**The greatest 3-digit number divisible by 4, 5, and 6 is 960.**

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