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Among the choices, find the sum of all three-digit natural numbers which are divisible by 13.

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41,456

40,231

37,566

37,674

answer is D.

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Detailed Solution

We have to find the sum of all three-digit natural numbers which are divisible by 13.
We know that the three-digit numbers that are completely divisible by 13 are:
104, 117, 130, 143,… 988
The above numbers are in the form of an AP.
So,

a 1 = 104 d = 13 a n = 988

The nth term of an AP is given by,

a n = a + (n − 1) d

.
To find the value of n, substitute the known values in the formula.

104 + (n − 1) × 13 = 988 104 + 13 n − 13 = 988 104 + 13 n = 1001 13 n = 1001 − 104 n = 69

There are 69 three digit numbers that are divisible by 13.
The formula that gives the sum of the n terms is,

S n = n 2 [a + l]

, where l is the last term.
Substitute the known values in the above formula.

S 29 = 692 [104 + 988]

= 292 (1092) = 37, 674

The sum of the three-digit numbers divisible by 13 is 37,674.
Hence, option 4) is correct.

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