Find the highest number less than 1900 that is divisible by each of the numbers 2, 3, 4, 5, 6 and 7.

Answer

To find the highest number less than 1900 that is divisible by each of the numbers 2, 3, 4, 5, 6, and 7, we will follow these steps:

### Step 1: Find the LCM of the numbers
We need to find the least common multiple (LCM) of the numbers 2, 3, 4, 5, 6, and 7.

- The prime factorization of each number is:
- 2 = 2
- 3 = 3
- 4 = 2^2
- 5 = 5
- 6 = 2 × 3
- 7 = 7

- To find the LCM, we take the highest power of each prime number that appears in these factorizations:
- The highest power of 2 is 2^2 (from 4)
- The highest power of 3 is 3^1 (from 3 and 6)
- The highest power of 5 is 5^1 (from 5)
- The highest power of 7 is 7^1 (from 7)

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 \]

Calculating this step by step:
- \( 4 \times 3 = 12 \)
- \( 12 \times 5 = 60 \)
- \( 60 \times 7 = 420 \)

So, the LCM of 2, 3, 4, 5, 6, and 7 is **420**.

### Step 2: Find the highest multiple of LCM less than 1900
Now we need to find the highest multiple of 420 that is less than 1900.

- To do this, we divide 1900 by 420 and take the integer part of the result:
\[ \frac{1900}{420} \approx 4.5238 \]
The integer part is **4**.

### Step 3: Calculate the highest multiple
Now we multiply the LCM by this integer:
\[ 420 \times 4 = 1680 \]

### Conclusion
The highest number less than 1900 that is divisible by 2, 3, 4, 5, 6, and 7 is **1680**.

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