How many numbers between 100 and 400 are divisible by 6?

There are 50 numbers between 100 and 400 that are divisible by 6.

To determine this, we can follow these steps:

1. Find the first multiple of 6 greater than 100: Divide 100 by 6, which gives approximately 16.67. The next whole number is 17. Therefore, the first multiple of 6 in the range is 17 * 6 = 102.

2. Find the last multiple of 6 less than 400: Divide 400 by 6, which gives approximately 66.67. The preceding whole number is 66. Therefore, the last multiple of 6 in the range is 66 * 6 = 396.

3. Create an arithmetic sequence: The numbers divisible by 6 between 100 and 400 form an arithmetic sequence: 102, 108, 114, ..., 396. The common difference is 6.

4. Use the arithmetic sequence formula to find the number of terms: The formula for the nth term of an arithmetic sequence is: a_n = a₁ + (n - 1)d, where a_n is the last term, a₁ is the first term, n is the number of terms, and d is the common difference.

   • 396 = 102 + (n - 1)6

5. Solve for n:

   • 396 - 102 = (n - 1)6
   • 294 = (n - 1)6
   • 294 / 6 = n - 1
   • 49 = n - 1
   • n = 49 + 1
   • n = 50

Therefore, there are 50 numbers between 100 and 400 that are divisible by 6.