How Many Numbers Between 500 and 1,000 Are Divisible by 13?

There are 38 numbers between 500 and 1000 that are divisible by 13.

Understanding the Calculation

To find the number of multiples of 13 within a given range, we can use the concept of arithmetic progressions. The multiples of 13 form an arithmetic sequence with a common difference of 13.

1. Find the first multiple: The smallest multiple of 13 greater than or equal to 500 is 507 (13 x 39).

2. Find the last multiple: The largest multiple of 13 less than or equal to 1000 is 988 (13 x 76).

3. Calculate the number of terms: We can use the formula for the nth term of an arithmetic progression: a_n = a₁ + (n-1)d, where a_n is the nth term, a₁ is the first term, n is the number of terms, and d is the common difference. In this case:

   • a_n = 988
   • a₁ = 507
   • d = 13

   Solving for n: 988 = 507 + (n-1)13 => n = 38

Therefore, there are 38 numbers between 500 and 1000 that are divisible by 13. This is consistent across multiple sources cited. For example, this Toppr.com answer directly states there are 38 such numbers. Similarly, other sources like Testbook.com and Testbook.com confirm this result.