Question

How many numbers between 500 and 1000 are divisible by 13?

Answer 1

Hint: You can form an AP of numbers between 500 and 1000 that are divisible by 13. Then use the formula for the n^th term of the AP given as \[{t_n} = a + (n - 1)d\] to form an inequality and find the integer value of n.

Complete step-by-step solution -
We need to find the number of numbers between 500 and 1000 that is divisible by 13. Now, we divide 500 by 13 to find the value of the first term that is divisible by 13. Hence, we have:
\[\dfrac{{500}}{{13}} = 38.46\]
Hence the first term is obtained as below:
\[a = 39 \times 13\]
\[a = 507\]
An arithmetic progression (AP) is a sequence of numbers whose consecutive terms differ by a constant number. This constant number is called the common ratio.
Hence, the numbers between 500 and 1000 that are divisible by 13 also form an AP with the first term 507 and the common difference 13.
The last term of the AP is the highest number in the AP that is less than or equal to 1000.
The formula for the nth term of the AP with the first term a and the common difference d is given as follows:
\[{t_n} = a + (n - 1)d\]
Hence, we have as follows:
\[{t_n} \leqslant 1000\]
\[507 + (n - 1)(13) \leqslant 1000\]
Simplifying, we have:
\[507 + 13n - 13 \leqslant 1000\]
\[494 + 13n \leqslant 1000\]
Let us take 494 to the other side of the inequality. Then, we have as follows:
\[13n \leqslant 1000 - 494\]
\[13n \leqslant 506\]
Solving for n, we have:
\[n \leqslant \dfrac{{506}}{{13}}\]
\[n \leqslant 38.9\]
Hence, the largest integer value for n is 38.
\[n = 38\]
Hence, the number of numbers between 500 and 1000 that is divisible by 13 is 38.

Note: You can also find the number of numbers between 500 and 1000 that is divisible by 13 by dividing the difference between 500 and 1000 by 13 and taking the quotient, you will get the same answer.