Question

The number of integers between 25 and 125, which are divisible by 6, is

• A. 15
• B. 14
• C. 16
• D. 18

Hint:

Use $\frac{last - first}{step} + 1$ to count evenly spaced numbers.

Answer 1

The Correct Option is C

To determine how many integers between 25 and 125 are divisible by 6, we need to find the range of numbers in this interval that can be evenly divided by 6. Let's solve this step-by-step:

1. Identify the smallest integer greater than or equal to 25 that is divisible by 6:
   1. Divide 25 by 6: \(\frac{25}{6} \approx 4.1667\).
   2. The next whole number is 5, so the smallest integer = \(5 \times 6 = 30\).
2. Identify the largest integer less than or equal to 125 that is divisible by 6:
   1. Divide 125 by 6: \(\frac{125}{6} \approx 20.8333\).
   2. The largest whole number less than this quotient is 20, so the largest integer = \(20 \times 6 = 120\).
3. Determine the total number of integers divisible by 6 from 30 to 120:
   1. Use the arithmetic sequence formula for integers: The general formula for the \(n\)-th term of an arithmetic sequence is \(T_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.
   2. Here, \(a = 30\), \(d = 6\), and the last term is 120.
   3. Equating \(T_n = 120\), we get: \(120 = 30 + (n-1) \times 6\).
   4. Simplifying: \(120 - 30 = (n-1) \times 6\)
   5. 90 = 6(n-1)
   6. 15 = n-1
   7. \(n = 16\)

Thus, there are 16 integers between 25 and 125 that are divisible by 6. Therefore, the correct answer is 16.