How many three-digit natural numbers are divisible by 7? (CBSE 2013)

All three-digit natural numbers which are divisible by 7 are as follows-

105, 112, 119, …., 994

The above Series of numbers is an A.P as the difference between the consecutive terms is constant.

Let the first term and common difference of given AP be a and d respectively.

first term(a) = 105 and,

common difference(d) = (n+1)th term - nth term = 112-105 = 7

last term or nth term(a_n) =994

Let the total no. of terms in above A.P be n.

∴ a_n = a+(n-1)×d

⇒ 994 = 105+(n-1)×7

⇒ 889 = 7n-7

⇒ 7n = 896

∴ n = 128

Thus, the no. of all three-digit numbers divisible by 7 = 128.