Q) How many two digits numbers are divisible by 3?

Ans: Here, we need to count all two-digit numbers which are divisible by 3.

Step 1: Now we know that the smallest two-digit number is 10.

∴ the first two digits number, which is also a multiple of 3, is 12.

The largest two-digit number is 99, and it is also multiple of 3.

∴ We need to count all two-digit numbers which are multiples of 3.

Step 2: Since starting with 12, these numbers will increase by 3,

∴ we can write them as an AP: 12, 15, 18, ……, 99

Step 3: This AP has, first term, a = 12

and common difference, d = 3

and last term, L = 99

Let’s consider this last term is n^th term of this AP.

We know that the value of n^th term of an AP is given by:

T_n = a + (n – 1) d

By substitute above values, we get:

∴ 99 = 12 + (n – 1) (3)

∴ 99 = 12 + 3 n – 3

∴ 99 = 9 + 3n

∴ 3 n = 99 – 9 = 90

∴ n  = = 30

Therefore, there are 30 two-digit numbers which are divisible by 3.

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