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

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answer is A.

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Detailed Solution

Concept- Then use the formula for the nth term of the AP given as

t_{n} = a + (n - 1) d

to make an inequality and find the integer value of

n

.
Now, we divide 500 by 13 to seek out the value of the first term that is divisible by 13. Hence, we have:

\frac{500}{13} = 38.46

So, the

1^{st}

term is obtained as below:

a = 39 \times 13

a = 507

A progression (AP) is a sequence of numbers whose consecutive terms differ by a constant number. This constant number is named the common ratio.
The formula for the nth term of the AP with the

1^{st}

term a and the common difference d is given as follows:

t_{n} = a + (n - 1) d

Then, we have as follows:

t_{n} ⩽ 1000

507 + (n - 1) (13) ⩽ 1000

Simplifying, we have:

507 + 13 n - 13 ⩽ 1000

494 + 13 n ⩽ 1000

Then, we have as follows:

13 n ⩽ 1000 - 494

13 n ⩽ 506

Solving for n, we have:

n ⩽ \frac{506}{13}

n ⩽ 38.9

So, the most important integer value for n is 38.

n = 38

Hence, the right option is 1.

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