How many 2-digit positive integers are divisible by 4 or 9?

Solution(By Examveda Team)

Number of 2-digit positive integers divisible by 4
The smallest 2-digit positive integer divisible by 4 is 12. The largest 2-digit positive integer divisible by 4 is 96.
All the 2-digit positive integers are terms of an Arithmetic progression with 12 being the first term and 96 being the last term.
The common difference is 4.
The n^th term a_n = a₁ + (n - 1)d, where a₁ is the first term, 'n' number of terms and 'd' the common difference.
So, 96 = 12 + (n - 1) × 4
84 = (n - 1) × 4
Or (n - 1) = 21
Hence, n = 22
i.e., there are 22 2-digit positive integers that are divisible by 4.

Number of 2-digit positive integers divisible by 9
The smallest 2-digit positive integer divisible by 9 is 18. The largest 2-digit positive integer divisible by 9 is 99.
All the 2-digit positive integers are terms of an Arithmetic progression with 18 being the first term and 99 being the last term.
The common difference is 9
The nth term an = a1 + (n - 1)d, where a1 is the first term, 'n' number of terms and 'd' the common difference.
So, 99 = 18 + (n - 1) × 9
Or 81 = (n - 1) × 9
Or (n - 1) = 9
Hence, n = 10
i.e., there are 10 2-digit positive integers that are divisible by 9.
Removing double count of numbers divisible by 4 and 9
Numbers such as 36 and 72 are multiples of both 4 and 9 and have therefore been counted in both the groups.
There are 2 such numbers.
Hence, number of 2-digit positive integers divisible by 4 or 9
= Number of 2-digit positive integers divisible by 4 + Number of 2-digit positive integers divisible by 4 - Number of 2-digit positive integers divisible by 4 and 9
= 22 + 10 - 2
= 30