What is the sum of the numbers between 300 and 500 such that when they are divided by 6, 12 and 16, it leaves no remainder?

Options

A. 1586
B. 1632
C. 1764
D. 1618

Answer

To find the sum of the numbers between 300 and 500 that are divisible by 6, 12, and 16, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 6, 12, and 16.

- The prime factorization of the numbers is:
- 6 = 2 × 3
- 12 = 2² × 3
- 16 = 2⁴

To find the LCM, we take the highest power of each prime:
- For 2, the highest power is 2⁴ (from 16).
- For 3, the highest power is 3¹ (from 6 or 12).

Thus, the LCM is:
\[ \text{LCM} = 2^4 \times 3^1 = 16 \times 3 = 48 \]

### Step 2: Identify the Range
We are looking for numbers between 300 and 500 that are divisible by 48.

### Step 3: Find the First and Last Multiples of 48 in the Range
- The first multiple of 48 greater than or equal to 300 can be found by calculating:
\[ \text{First multiple} = \lceil \frac{300}{48} \rceil \times 48 \]
\[ \frac{300}{48} \approx 6.25 \Rightarrow \lceil 6.25 \rceil = 7 \]
\[ \text{First multiple} = 7 \times 48 = 336 \]

- The last multiple of 48 less than or equal to 500 can be found by calculating:
\[ \text{Last multiple} = \lfloor \frac{500}{48} \rfloor \times 48 \]
\[ \frac{500}{48} \approx 10.42 \Rightarrow \lfloor 10.42 \rfloor = 10 \]
\[ \text{Last multiple} = 10 \times 48 = 480 \]

### Step 4: List the Multiples of 48 Between 336 and 480
The multiples of 48 in this range are:
- 336
- 384
- 432
- 480

### Step 5: Calculate the Sum of These Multiples
Now, we will sum these numbers:
\[ 336 + 384 + 432 + 480 \]

Calculating step by step:
1. \( 336 + 384 = 720 \)
2. \( 720 + 432 = 1152 \)
3. \( 1152 + 480 = 1632 \)

Thus, the sum of the numbers between 300 and 500 that are divisible by 6, 12, and 16 is **1632**.

### Final Answer
The sum is **1632**.

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