Bhavagna celebrates her birthday every year. The number of chocolates that she distributes every year is the greatest 3- digit common multiple of all the prime factors of the number representing that year. How many chocolates should bhavagna distribute in the year 2016 ?

Explanation

To find out how many chocolates Bhavna should distribute in the year 2016, we need to follow these steps:

### Step 1: Find the Prime Factors of 2016
To find the prime factors of 2016, we can start dividing by the smallest prime numbers.

1. **Divide by 2**:
- 2016 ÷ 2 = 1008
- 1008 ÷ 2 = 504
- 504 ÷ 2 = 252
- 252 ÷ 2 = 126
- 126 ÷ 2 = 63 (63 is not divisible by 2 anymore)

2. **Divide by 3**:
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7 (7 is a prime number)

So, the prime factorization of 2016 is:
\[ 2016 = 2^5 \times 3^2 \times 7^1 \]
The prime factors are 2, 3, and 7.

### Step 2: Find the Least Common Multiple (LCM) of the Prime Factors
Next, we need to find the LCM of the prime factors 2, 3, and 7. Since they are all prime numbers, the LCM will be their product:
\[ LCM = 2 \times 3 \times 7 = 42 \]

### Step 3: Find the Greatest 3-Digit Multiple of 42
Now, we need to find the greatest 3-digit multiple of 42. The largest 3-digit number is 999.

1. **Divide 999 by 42**:
- 999 ÷ 42 ≈ 23.7857 (take the integer part, which is 23)

2. **Multiply back to find the greatest multiple**:
- 42 × 23 = 966

Thus, the greatest 3-digit multiple of 42 is 966.

### Conclusion
Therefore, Bhavna should distribute **966 chocolates** in the year 2016.

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