How many numbers of two digit are divisible by 3?
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
∴ a1 = a = 12
Next two digit number which is divisible by 3 is 15
∴ a2 = 15
Largest two digit number which is divisible by 3 is 99
∴ an = 99
⇒ A.P. is 12, 15,………,99
We know, an = a + (n – 1)d where a is first term or a and d is common difference and n is any natural number
⇒ a1 = 12, a2 = 15 and an = 99
Common difference, d1 = a2 – a1 = 15 – 12 = 3
Now, an = a1 + (n – 1)d
⇒ an = 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