Counting Numbers Not Divisible by 2, 3, or 5 Between 100 and 200

In this article, we will explore how to determine the count of integers between 100 and 200 (inclusive) that are not divisible by 2, 3, or 5. We will employ the principle of complementary counting and the inclusion-exclusion principle to find the solution. This method involves counting all numbers in the range and subtracting those divisible by 2, 3, or 5.

Step 1: Count the Total Numbers from 100 to 200

The total count of numbers between 100 and 200 inclusive is:

200 - 100 1 101

Step 2: Use the Inclusion-Exclusion Principle to Count Divisibility

Let:

A represent the set of numbers divisible by 2. B represent the set of numbers divisible by 3. C represent the set of numbers divisible by 5.

Our goal is to calculate the size of the union of sets A, B, and C (numbers divisible by at least one of 2, 3, or 5).

Using the inclusion-exclusion principle:

A ∪ B ∪ C A B C - A ∩ B - A ∩ C - B ∩ C A ∩ B ∩ C

Step 3: Calculate Each Component

Calculate A: Numbers Divisible by 2

The sequence of numbers divisible by 2 from 100 to 200 is:

100, 102, 104, ..., 200

An arithmetic sequence where:

First term a 100 Last term l 200 Common difference d 2

The number of terms n can be calculated as:

n (l - a) / d 1 (200 - 100) / 2 1 51

Calculate B: Numbers Divisible by 3

The sequence of numbers divisible by 3 from 100 to 200 is:

102, 105, 108, ..., 198

An arithmetic sequence where:

First term a 102 Last term l 198 Common difference d 3

The number of terms n can be calculated as:

n (l - a) / d 1 (198 - 102) / 3 1 33

Calculate C: Numbers Divisible by 5

The sequence of numbers divisible by 5 from 100 to 200 is:

100, 105, 110, ..., 200

An arithmetic sequence where:

First term a 100 Last term l 200 Common difference d 5

The number of terms n can be calculated as:

n (l - a) / d 1 (200 - 100) / 5 1 21

Calculate A ∩ B: Numbers Divisible by 6 (LCM of 2 and 3)

The sequence is:

102, 108, 114, ..., 198

An arithmetic sequence where:

First term a 102 Last term l 198 Common difference d 6

The number of terms n can be calculated as:

n (l - a) / d 1 (198 - 102) / 6 1 17

Calculate A ∩ C: Numbers Divisible by 10 (LCM of 2 and 5)

The sequence is:

100, 110, 120, ..., 200

An arithmetic sequence where:

First term a 100 Last term l 200 Common difference d 10

The number of terms n can be calculated as:

n (l - a) / d 1 (200 - 100) / 10 1 11

Calculate B ∩ C: Numbers Divisible by 15 (LCM of 3 and 5)

The sequence is:

105, 120, 135, ..., 195

An arithmetic sequence where:

First term a 105 Last term l 195 Common difference d 15

The number of terms n can be calculated as:

n (l - a) / d 1 (195 - 105) / 15 1 7

Calculate A ∩ B ∩ C: Numbers Divisible by 30 (LCM of 2, 3, and 5)

The sequence is:

120, 150, 180

An arithmetic sequence where:

First term a 120 Last term l 180 Common difference d 30

The number of terms n can be calculated as:

n (l - a) / d 1 (180 - 120) / 30 1 3

Step 4: Plug Values into the Inclusion-Exclusion Formula

A ∪ B ∪ C 51 33 21 - 17 - 11 - 7 3 73

Step 5: Calculate the Count of Numbers Not Divisible by 2, 3, or 5

The count of numbers not divisible by 2, 3, or 5 is:

101 - 73 28

Final Answer

There are 28 numbers between 100 and 200 that are not divisible by 2, 3, or 5.