Question: How many positive integers less than 1000 are divisible by both 4 and 6 but not by 24? - Sterling Industries
How Many Positive Integers Less Than 1000 Are Divisible by Both 4 and 6 but Not by 24?
Uncover the surprising logic behind this common number puzzle
How Many Positive Integers Less Than 1000 Are Divisible by Both 4 and 6 but Not by 24?
Uncover the surprising logic behind this common number puzzle
In a world flooded with data and digital curiosity, one seemingly simple question sparks quiet fascination: How many positive integers less than 1000 are divisible by both 4 and 6 but not by 24? At first glance, it’s a math puzzle—yet it reveals patterns that resonate across time and technology trends. Many readers ask it during educational dives, budget planning for infrastructure, or while exploring patterns in digital systems. With the rise of smart algorithms and automated number analysis, this inquiry reflects broader interests in efficiency, categorization, and predictable outcomes in a complex world.
Why This Question Is Rising in Online Interest
Understanding the Context
Curiosity about divisibility rules isn’t just academic—it taps into deeper digital habits. As users interact with tools powered by search intent, questions like this often surface when exploring number theory simplifications, programming logic, or backend system requirements. The requirement to exclude multiples of 24 adds a layer of cognitive challenge, engaging problem-solving instincts. In the U.S., where STEM interest and data literacy grow steadily, such questions align with everyday learning moments—whether for students, educators, or tech-savvy professionals.
How the Calculation Actually Works
To count numbers less than 1000 divisible by both 4 and 6 but not by 24, start by finding numbers divisible by their least common multiple (LCM). The LCM of 4 and 6 is 12. So we first count multiples of 12 below 1000:
- The largest multiple of 12 below 1000 is 12 × 83 = 996
- Total = 83 such numbers
Key Insights
Now, exclude those also divisible by 24—the common multiple that eliminates uniqueness in the set.
Multiples of 24 below 1000:
- 24 × 41 = 984
- Total = 41 numbers divisible by 24
These 41 numbers are already included in the 83 multiples of 12, so subtract to preserve exclusivity:
83 – 41 = 42
Thus, there are exactly 42 positive integers less than 1000 divisible by
