Find the greatest 3-digit number which is common multiple of 6,15 and 21.

Explanation

To find the greatest 3-digit number that is a common multiple of 6, 15, and 21, we can follow these steps:

### Step 1: Find the LCM of 6, 15, and 21
To find the least common multiple (LCM), we will first find the prime factorization of each number:

- **6** = 2 × 3
- **15** = 3 × 5
- **21** = 3 × 7

Now, we take the highest power of each prime factor that appears in these factorizations:

- The prime factors are 2, 3, 5, and 7.
- The highest powers are:
- 2^1 from 6
- 3^1 from 6, 15, and 21
- 5^1 from 15
- 7^1 from 21

Now, we can calculate the LCM:

\[ \text{LCM} = 2^1 \times 3^1 \times 5^1 \times 7^1 = 2 \times 3 \times 5 \times 7 \]

Calculating this step-by-step:

1. \(2 \times 3 = 6\)
2. \(6 \times 5 = 30\)
3. \(30 \times 7 = 210\)

So, the LCM of 6, 15, and 21 is **210**.

### Step 2: Find the greatest 3-digit multiple of 210
Now, we need to find the greatest 3-digit number that is a multiple of 210. The largest 3-digit number is 999.

To find the greatest multiple of 210 that is less than or equal to 999, we can divide 999 by 210 and take the integer part of the result:

\[ 999 \div 210 \approx 4.757 \]

Taking the integer part, we get **4**. Now, we multiply 210 by 4:

\[ 210 \times 4 = 840 \]

### Conclusion
Thus, the greatest 3-digit number which is a common multiple of 6, 15, and 21 is **840**.

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