Understanding the Problem: Natural Numbers Not Divisible by Multiple Factors

The question asks us to find the count of natural numbers between 1 and 100 (inclusive) that are not divisible by any of the given numbers: 2, 3, 5, 7, and 9. This means we are looking for numbers that are simultaneously not divisible by 2, not divisible by 3, not divisible by 5, not divisible by 7, AND not divisible by 9.

Let's analyze the set of divisors: {2, 3, 5, 7, 9}. Notice that 9 is a multiple of 3. If a number is not divisible by 3, it cannot be divisible by 9. Therefore, the condition "not divisible by 3 AND not divisible by 9" is equivalent to simply "not divisible by 3". So, the list of unique divisibility conditions is simplified to "not divisible by 2, not divisible by 3, not divisible by 5, and not divisible by 7".

Thus, the problem is equivalent to finding the number of integers from 1 to 100 that are not divisible by any of the prime numbers 2, 3, 5, and 7.

Applying the Principle of Inclusion-Exclusion

To find the number of elements that satisfy none of the conditions (i.e., not divisible by any of 2, 3, 5, or 7), we can use the Principle of Inclusion-Exclusion. This principle helps us count the elements in the union of several sets. If we find the number of elements divisible by at least one of 2, 3, 5, or 7, we can subtract this from the total number of elements (100) to get the desired count.

Let N be the total number of natural numbers from 1 to 100, so N = 100.

Let A be the set of numbers in {1, ..., 100} divisible by 2.

Let B be the set of numbers in {1, ..., 100} divisible by 3.

Let C be the set of numbers in {1, ..., 100} divisible by 5.

Let D be the set of numbers in {1, ..., 100} divisible by 7.

We want to find N - $|A \cup B \cup C \cup D|$.

The formula for the Principle of Inclusion-Exclusion for four sets is:

$$|A \cup B \cup C \cup D| = \sum |Individual| - \sum |Pairwise Intersections| + \sum |Triple Intersections| - \sum |Quadruple Intersection|$$

Let's calculate each part.

Calculating Individual Divisibility Counts (Single Sets)

• Numbers divisible by 2: $|A| = \lfloor 100/2 \rfloor = 50$
• Numbers divisible by 3: $|B| = \lfloor 100/3 \rfloor = 33$
• Numbers divisible by 5: $|C| = \lfloor 100/5 \rfloor = 20$
• Numbers divisible by 7: $|D| = \lfloor 100/7 \rfloor = 14$

Sum of individual counts: $50 + 33 + 20 + 14 = 117$.

Calculating Pairwise Intersection Counts (Two Sets)

A number is divisible by two numbers if it is divisible by their least common multiple (LCM).

• Divisible by 2 and 3 (LCM(2,3)=6): $|A \cap B| = \lfloor 100/6 \rfloor = 16$
• Divisible by 2 and 5 (LCM(2,5)=10): $|A \cap C| = \lfloor 100/10 \rfloor = 10$
• Divisible by 2 and 7 (LCM(2,7)=14): $|A \cap D| = \lfloor 100/14 \rfloor = 7$
• Divisible by 3 and 5 (LCM(3,5)=15): $|B \cap C| = \lfloor 100/15 \rfloor = 6$
• Divisible by 3 and 7 (LCM(3,7)=21): $|B \cap D| = \lfloor 100/21 \rfloor = 4$
• Divisible by 5 and 7 (LCM(5,7)=35): $|C \cap D| = \lfloor 100/35 \rfloor = 2$

Sum of pairwise intersection counts: $16 + 10 + 7 + 6 + 4 + 2 = 45$.

Calculating Triple Intersection Counts (Three Sets)

• Divisible by 2, 3, and 5 (LCM(2,3,5)=30): $|A \cap B \cap C| = \lfloor 100/30 \rfloor = 3$
• Divisible by 2, 3, and 7 (LCM(2,3,7)=42): $|A \cap B \cap D| = \lfloor 100/42 \rfloor = 2$
• Divisible by 2, 5, and 7 (LCM(2,5,7)=70): $|A \cap C \cap D| = \lfloor 100/70 \rfloor = 1$
• Divisible by 3, 5, and 7 (LCM(3,5,7)=105): $|B \cap C \cap D| = \lfloor 100/105 \rfloor = 0$

Sum of triple intersection counts: $3 + 2 + 1 + 0 = 6$.

Calculating Quadruple Intersection Count (Four Sets)

• Divisible by 2, 3, 5, and 7 (LCM(2,3,5,7)=210): $|A \cap B \cap C \cap D| = \lfloor 100/210 \rfloor = 0$

Sum of quadruple intersection counts: $0$.

Applying Inclusion-Exclusion Formula to Find Union Size

Now substitute the calculated values into the formula:

$$|A \cup B \cup C \cup D| = (117) - (45) + (6) - (0)$$

$$|A \cup B \cup C \cup D| = 117 - 45 + 6 = 72 + 6 = 78$$

This means 78 numbers between 1 and 100 are divisible by at least one of 2, 3, 5, or 7.

Finding Numbers Not Divisible by Any of the Factors

The number of integers from 1 to 100 that are not divisible by any of 2, 3, 5, or 7 is the total number of integers minus the number of integers divisible by at least one of them.

Number of desired integers = Total numbers - $|A \cup B \cup C \cup D|$

Number of desired integers = $100 - 78 = 22$.

Therefore, there are 22 natural numbers among the first 100 that are not divisible by any one of 2, 3, 5, 7, and 9.

Revision Table: Divisibility Counts (1-100)

Additional Information: Key Concepts in Number Theory

Least Common Multiple (LCM): The LCM of two or more integers is the smallest positive integer that is a multiple of all the integers. When finding numbers divisible by multiple factors, the count is determined by the LCM of those factors within the given range.

Principle of Inclusion-Exclusion: This is a counting technique used to find the number of elements in the union of multiple sets. It accounts for elements that are counted multiple times in the simple sum of individual set sizes by subtracting the sizes of pairwise intersections, adding back the sizes of triple intersections, and so on.

For two sets A and B: $|A \cup B| = |A| + |B| - |A \cap B|$.

For three sets A, B, and C: $|A \cup B \cup C| = |A| + |B| + |C| - (|A \cap B| + |A \cap C| + |B \cap C|) + |A \cap B \cap C|$.

The pattern extends for more sets, alternating between addition and subtraction of the sums of the sizes of intersections of increasing numbers of sets.