Answer :

For finding total two-digit numbers which are divisible by 3, firstly we will make an A.P. of those two-digit numbers which are divisible by 3.

First two digit number which is divisible by 3 is 12

∴ a_{1} = a = 12

Next two digit number which is divisible by 3 is 15

∴ a_{2} = 15

Largest two digit number which is divisible by 3 is 99

∴ a_{n} = 99

⇒ A.P. is 12, 15,………,99

We know, a_{n} = a + (n – 1)d where a is first term or a and d is common difference and n is any natural number

⇒ a_{1} = 12, a_{2} = 15 and a_{n} = 99

Common difference, d_{1} = a_{2} – a_{1} = 15 – 12 = 3

Now, a_{n} = a_{1} + (n – 1)d

⇒ a_{n} = 12 + (n – 1)3

⇒ 99 = 12 + 3n – 3

⇒ 99 = 3n + 9

⇒ 99 – 9 = 3n

⇒ 90 = 3n

⇒ n = 30

Hence, there are total 30 two-digit numbers which are divisible by 3

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