Question

Find the number of integers between 1 and 1000 , including 1 and 1000 , that are not divisible by any of \(4,5,\) or 6.

Short answer

There are 534 integers between 1 and 1000 that are not divisible by 4, 5, or 6.

Step-by-step solution

Identify Total Numbers

First, determine the total number of integers between 1 and 1000, inclusive. This is simply the set of numbers from 1 to 1000, giving us 1000 integers.

Apply the Inclusion-Exclusion Principle

We will use the Inclusion-Exclusion Principle to find the count of numbers divisible by any of 4, 5, or 6. The principle helps in calculating the union of multiple sets where certain overlaps are considered.

Calculate Numbers Divisible by 4, 5, and 6

Find the number of integers divisible by each of the numbers 4, 5, and 6. Calculate these separately:- Divisible by 4: There are \[ \left\lfloor \frac{1000}{4} \right\rfloor = 250 \]integers.- Divisible by 5: There are \[ \left\lfloor \frac{1000}{5} \right\rfloor = 200 \]integers.- Divisible by 6: There are \[ \left\lfloor \frac{1000}{6} \right\rfloor = 166 \]integers.

Calculate Numbers Divisible by Pair Combinations

Now calculate numbers divisible by pairs of these numbers using their least common multiples (LCM):- Divisible by 20 (LCM of 4 and 5): \[ \left\lfloor \frac{1000}{20} \right\rfloor = 50 \]- Divisible by 12 (LCM of 4 and 6): \[ \left\lfloor \frac{1000}{12} \right\rfloor = 83 \]- Divisible by 30 (LCM of 5 and 6): \[ \left\lfloor \frac{1000}{30} \right\rfloor = 33 \]

Calculate Numbers Divisible by 4, 5, and 6 Together

Find the number of integers divisible by 4, 5, and 6 together using their LCM:- Divisible by 60 (LCM of 4, 5, and 6): \[ \left\lfloor \frac{1000}{60} \right\rfloor = 16 \]

Apply Inclusion-Exclusion to Find Union

Apply the Inclusion-Exclusion principle:\[|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|\]Substituting the values:\[250 + 200 + 166 - 50 - 83 - 33 + 16 = 466\]Thus, 466 numbers are divisible by at least one of 4, 5, or 6.

Subtract Divisible Numbers from Total

Finally, subtract the total numbers divisible by at least one of 4, 5, or 6 from the total number of integers:\[1000 - 466 = 534\]This gives us the number of integers between 1 and 1000 that are not divisible by 4, 5, or 6.

Key concepts

Divisibility

Divisibility is a fundamental concept in mathematics, referring to the ability of one integer to be divided by another without a remainder. When we say a number is divisible by another, it means that after dividing, what's left over, the remainder, is zero. For example, the number 20 is divisible by 5, because when you divide 20 by 5, you get 4 with no remainder.
Understanding divisibility is crucial in problems like determining how many numbers within a range have certain divisibility properties, as seen in our exercise. We use divisibility rules to quickly determine if one number is divisible by another without performing lengthy division. For instance, a number is divisible by 4 if the number formed by its last two digits is divisible by 4. Similarly, if a number ends in 0 or 5, it is divisible by 5, and if it’s even, it’s divisible by 2, which helps us in finding numbers divisible by 6 (since 6 is a product of 2 and 3).
In problems of inclusion-exclusion principle, identifying divisible numbers helps in setting apart specific categories of numbers. This concept becomes our base as we further explore least common multiples and counting integers within these constraints.

Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more integers is the smallest number that can be evenly divided by each of the integers. Finding the LCM is particularly important when combining divisibility conditions, as we must consider all possible overlaps.
To find the LCM of two numbers, you can list the multiples of each number until you find the smallest multiple they have in common. Another common method is using prime factorization, where you take the highest power of each prime that appears in any of the numbers and multiply them together.
For example, when you need to determine how many numbers are divisible by both 4 and 5, you would find their LCM, which is 20. Similarly, the LCM of 4, 5, and 6 is 60. These calculations are crucial in using the Inclusion-Exclusion Principle, as they help to calculate overlaps accurately, avoiding over-counting or missing numbers that meet multiple criteria in the set.

Counting integers

Counting integers that meet certain criteria, such as not being divisible by specific numbers, is a common problem in mathematics. The task involves systematic methods to ensure all eligible numbers are counted without duplicating those that fit none of the exclusion criteria.
In the exercise, the goal is to find how many integers from 1 to 1000 are not divisible by 4, 5, or 6. The solution starts by finding the total number of integers within the range, then using the Inclusion-Exclusion Principle to calculate those that are divisible by any of the given numbers. Once you know how many integers meet at least one of the divisibility conditions, you subtract this quantity from the total number of integers.

• This method ensures you accurately count integers of interest, making it possible to systematically evaluate different combinations and intersections of criteria.
• The systematic application of these steps is crucial in deriving the correct answer and understanding how these mathematical tools work together to solve complex counting problems.

Each step leverages the detailed calculation from divisibility numbers and LCM, culminating in a clear and complete count.