Counting Positive Integers Not Divisible by 3, 5, and 7 up to 6300
In this article, we will explore how to find the number of positive integers less than or equal to 6300 that are not divisible by 3, 5, and 7. We will utilize the principle of Inclusion-Exclusion to solve this problem. The method described will be explained in detail, along with a step-by-step breakdown of the calculation.
The Inclusion-Exclusion Principle
The principle of Inclusion-Exclusion is a useful tool in combinatorics for counting elements in the union of multiple sets. It allows us to find the size of the union of multiple sets by summing the sizes of the individual sets, subtracting the sizes of their intersections, and adding or subtracting the sizes of their larger intersections, alternating between addition and subtraction as we go.
Problem Overview
We are tasked with finding the number of positive integers up to 6300 that are not divisible by 3, 5, or 7. This is equivalent to finding the positive integers that are not in the union of sets A, B, and C, where:
A is the set of integers divisible by 3 B is the set of integers divisible by 5 C is the set of integers divisible by 7Step-by-Step Solution
We will use the Inclusion-Exclusion Principle to find the count of integers in the union of these sets and then subtract that from the total number of positive integers up to 6300.
Step 1: Counting Multiples of Individual Numbers
First, we calculate the number of integers up to 6300 that are multiples of 3, 5, and 7:
Count of multiples of 3: mathA leftlfloor frac{6300}{3} rightrfloor 2100/math Count of multiples of 5: mathB leftlfloor frac{6300}{5} rightrfloor 1260/math Count of multiples of 7: mathC leftlfloor frac{6300}{7} rightrfloor 900/mathStep 2: Counting Multiples of Pairs of Numbers
Next, we account for the integers that are multiples of pairs of these numbers:
Count of multiples of 15 (3 and 5): mathA cap B leftlfloor frac{6300}{15} rightrfloor 420/math Count of multiples of 21 (3 and 7): mathA cap C leftlfloor frac{6300}{21} rightrfloor 300/math Count of multiples of 35 (5 and 7): mathB cap C leftlfloor frac{6300}{35} rightrfloor 180/mathStep 3: Counting Multiples of All Three Numbers
Finally, we consider the integers that are multiples of all three numbers (105):
Count of multiples of 105 (3, 5, and 7): mathA cap B cap C leftlfloor frac{6300}{105} rightrfloor 60/mathStep 4: Applying Inclusion-Exclusion Principle
Using the principle of Inclusion-Exclusion, we find the total number of integers in the union of sets A, B, and C:
mathA cup B cup C A B C - (A cap B) - (A cap C) - (B cap C) (A cap B cap C)/math
mathA cup B cup C 2100 1260 900 - 420 - 300 - 180 60 3530/math
Step 5: Finding the Count of Integers Not Divisible by 3, 5, or 7
Since the total number of positive integers up to 6300 is 6300, the count of integers not divisible by 3, 5, or 7 is:
6300 - 3530 2770
A Verification Method
To verify our solution, we can use another method. The least common multiple (LCM) of 3, 5, and 7 is 105. Dividing 6300 by 105 gives us 60 sets of 105. We can then calculate the integers not divisible by 3, 5, or 7 within each set and sum them up.
Two-thirds of the set (105) are not divisible by 3, which is 105 * (2/3) 70. Out of these 70 integers, four-fifths are not divisible by 5, which is 70 * (4/5) 56. Finally, six-sevenths of these remaining 56 integers are not divisible by 7, which is 56 * (6/7) 48.
Therefore, the total number of integers not divisible by 3, 5, or 7 is 60 sets of 48, which is 60 * 48 2880. This is also equivalent to the calculation using the total number of integers up to 6300: 6300 - (2/3) * (4/5) * (6/7) * 6300 2880.
In conclusion, using the principle of Inclusion-Exclusion, we have determined that there are 2770 positive integers less than or equal to 6300 that are not divisible by 3, 5, or 7.