There are 128 three-digit natural numbers divisible by 7.
Understanding the Calculation
To find the number of three-digit numbers divisible by 7, we can use the concept of arithmetic progressions. The smallest three-digit number divisible by 7 is 105 (7 x 15), and the largest is 994 (7 x 142). The common difference between consecutive three-digit multiples of 7 is, of course, 7.
Therefore, we have an arithmetic progression with:
- First term (a): 105
- Last term (l): 994
- Common difference (d): 7
The formula to find the number of terms (n) in an arithmetic progression is:
n = (l - a) / d + 1
Substituting our values:
n = (994 - 105) / 7 + 1 = 889 / 7 + 1 = 127 + 1 = 128
Hence, there are 128 three-digit numbers divisible by 7. This aligns with the information provided in several sources stating the answer is 128. For example, one source directly states that "Hence, there are 128 three-digit numbers divisible by 7."
Practical Application
This type of calculation is useful in various scenarios, such as:
- Programming: Determining the number of iterations in a loop processing numbers divisible by 7.
- Number Theory: Solving problems related to divisibility and arithmetic progressions.
- Data Analysis: Filtering datasets based on divisibility criteria.
