What is the Total Number of Integers Between 100 and 1000 Divisible by 7?

There are 128 integers between 100 and 1000 that are divisible by 7.

Understanding the Calculation

This problem can be solved using arithmetic sequences. The integers divisible by 7 form an arithmetic progression (AP) where the first term is 105 (the smallest multiple of 7 greater than 100) and the common difference is 7. The last term is 994 (the largest multiple of 7 less than 1000).

The general formula for the nth term of an arithmetic progression is: an = a + (n-1)d, where:

• an is the nth term
• a is the first term
• n is the number of terms
• d is the common difference

In this case:

• an = 994
• a = 105
• d = 7

We need to solve for n:

994 = 105 + (n-1)7

Solving for 'n', we find that there are 128 terms in the sequence. Therefore, there are 128 integers between 100 and 1000 divisible by 7. Multiple sources (Toppr, Quora, Vedantu, Testbook, and others) confirm this result. A shortcut method is to calculate 1000/7 - 100/7 which approximates to 142 - 14 = 128.

Conclusion