How many number of two-digit positive integer < than 100, which are not…

Solution: Count two-digit integers and use inclusion–exclusion to remove those divisible by 2, 3, or 5.

• Total two-digit integers (10 to 99): 90.

• Count divisible by 2: 45.

• Count divisible by 3: 30.

• Count divisible by 5: 18.

• Pairwise overlaps: divisible by 6 (2 and 3): 15; by 10 (2 and 5): 9; by 15 (3 and 5): 6.

• Triple overlap (divisible by 30): 3.

• By inclusion–exclusion, numbers divisible by at least one of 2, 3, 5 = 45 + 30 + 18 − (15 + 9 + 6) + 3 = 66.

• Thus numbers not divisible by 2, 3, or 5 = 90 − 66 = 24.

Answer: 24