How many two-digit numbers are divisible by 3 ?

Two digits no. divisible by 3 are,

12, 15, 18 ………, 99.

The above series is an A.P. with, a = 12 and d = 15 - 12 = 3 and last term = 99.

Let total no. of terms be n,

∴ a_n = a + (n - 1)d

⇒ 99 = 12 + (n - 1)3

⇒ 99 = 12 + 3n - 3

⇒ 99 = 9 + 3n

⇒ 99 - 9 = 3n

⇒ 90 = 3n

⇒ n = 30.

Hence, two-digit numbers divisible by 3 are 30.