Numbers Not Divisible By 2 Or 3: 20-110
Hey math whizzes and number crunchers! Today, we're diving deep into a classic number theory puzzle that's super common in math discussions and even pops up in competitive exams. We're going to figure out how many natural numbers greater than 20 and less than 110 are not divisible by either 2 or 3. This might sound a bit tricky, but trust me, guys, once you break it down, it's totally manageable and actually pretty cool. We'll be using some fundamental principles of counting and set theory, specifically the Inclusion-Exclusion Principle, to crack this one. So, grab your thinking caps, maybe a pen and paper, and let's get this mathematical party started! We're not just aiming for the answer; we want to understand the why behind it, making sure you can tackle similar problems with confidence. Get ready to flex those brain muscles and explore the fascinating world of numbers that dance to their own tune, avoiding the common beats of divisibility by two and three.
Understanding the Problem: Breaking It Down
Alright, let's get crystal clear on what we're looking for. The problem asks us to count natural numbers that fit a specific range and meet certain divisibility criteria. The range is greater than 20 and less than 110. This means we're considering numbers from 21 up to 109, inclusive. So, the set of numbers we're working with is {21, 22, 23, ..., 109}. Now, the crucial part: we want numbers that are not divisible by 2 AND not divisible by 3. This means we're excluding all even numbers (multiples of 2) and all multiples of 3 from our count. It's like trying to find the black sheep in a flock of white ones, but with numbers!
To tackle this, we can use a strategy called the Inclusion-Exclusion Principle. It's a super handy tool for counting elements in the union of multiple sets. In our case, we can start with the total number of integers in our range and then subtract the numbers that are divisible by 2 and the numbers that are divisible by 3. But wait! If we just subtract them, we'll end up subtracting the numbers divisible by both 2 and 3 (which are the multiples of 6) twice. So, we need to add them back in once. This principle helps us avoid overcounting or undercounting.
First, let's figure out the total count of natural numbers between 20 and 110 (exclusive of 20 and 110, so 21 to 109). The formula for the number of integers in a range [a, b] is b - a + 1. So, for our range [21, 109], the total number of integers is 109 - 21 + 1 = 89. We have 89 numbers to play with.
Next, we need to find out how many of these 89 numbers are divisible by 2, how many are divisible by 3, and how many are divisible by both 2 and 3 (i.e., divisible by 6). The numbers that are divisible by 2 or 3 are the ones we want to exclude. The numbers we want are those that are left after we've removed all multiples of 2 and all multiples of 3.
So, the core idea is: Total Numbers - (Numbers divisible by 2 OR Numbers divisible by 3) = Numbers NOT divisible by 2 AND NOT divisible by 3. And using Inclusion-Exclusion, Numbers divisible by 2 OR Numbers divisible by 3 = (Numbers divisible by 2) + (Numbers divisible by 3) - (Numbers divisible by 6). Phew! Sounds like a plan, right? Let's get calculating!
Counting the Multiples: Step-by-Step Calculation
Alright guys, let's get down to the nitty-gritty calculations. We've established our range of natural numbers is from 21 to 109, giving us a total of 89 numbers. Now, we need to count the multiples within this range for divisors 2, 3, and 6.
1. Numbers Divisible by 2:
We need to find the multiples of 2 between 21 and 109. The first multiple of 2 greater than 20 is 22. The last multiple of 2 less than 110 is 108. To count them, we can use the formula: floor(last_multiple / divisor) - floor((first_number - 1) / divisor).
So, for numbers up to 109 divisible by 2: floor(109 / 2) = 54. This counts multiples of 2 from 1 up to 109.
For numbers up to 20 divisible by 2: floor(20 / 2) = 10. This counts multiples of 2 from 1 up to 20.
The number of multiples of 2 between 21 and 109 is 54 - 10 = 44. So, there are 44 numbers divisible by 2 in our range.
2. Numbers Divisible by 3: Now, let's find the multiples of 3 between 21 and 109. The first multiple of 3 in our range is 21 itself. The last multiple of 3 less than 110 is 108.
For numbers up to 109 divisible by 3: floor(109 / 3) = 36.
For numbers up to 20 divisible by 3: floor(20 / 3) = 6.
The number of multiples of 3 between 21 and 109 is 36 - 6 = 30. So, there are 30 numbers divisible by 3 in our range.
3. Numbers Divisible by 6 (Both 2 and 3): These are the multiples of the least common multiple of 2 and 3, which is 6. We need to find the multiples of 6 between 21 and 109. The first multiple of 6 in our range is 24. The last multiple of 6 less than 110 is 108.
For numbers up to 109 divisible by 6: floor(109 / 6) = 18.
For numbers up to 20 divisible by 6: floor(20 / 6) = 3.
The number of multiples of 6 between 21 and 109 is 18 - 3 = 15. So, there are 15 numbers divisible by 6 in our range.
Now we can use the Inclusion-Exclusion Principle to find the number of integers divisible by 2 OR 3:
Numbers (divisible by 2 or 3) = Numbers (divisible by 2) + Numbers (divisible by 3) - Numbers (divisible by 6)
Numbers (divisible by 2 or 3) = 44 + 30 - 15 = 74 - 15 = 59.
So, there are 59 numbers between 21 and 109 (inclusive) that are divisible by either 2 or 3 (or both). Pretty neat, huh?
The Final Count: Finding the Numbers We Want
We're almost there, folks! We've done the heavy lifting with the calculations. We know the total number of integers in our range, and we know how many of them are divisible by 2 or 3. The question asks for the numbers that are not divisible by 2 AND not divisible by 3. This is exactly what's left after we remove all the numbers divisible by 2 or 3 from our total set.
So, the final calculation is straightforward:
Numbers (not divisible by 2 AND not divisible by 3) = Total Numbers - Numbers (divisible by 2 or 3)
We calculated the total number of integers in the range [21, 109] as 89.
We calculated the number of integers divisible by 2 or 3 in that range using the Inclusion-Exclusion Principle as 59.
Therefore, the number of natural numbers greater than 20 and less than 110 that are not divisible by 2 or 3 is:
89 - 59 = 40.
And there you have it! The answer is 40. Isn't that cool? We found 40 numbers in that specific range that avoid being multiples of 2 and 3. These are the numbers whose prime factors do not include 2 or 3. Think of numbers like 23, 25, 29, 31, 35, 37, and so on. They're the ones that kind of stand out by not being