Question

Determine how many numbers between 1 and \(21,000,000,000,\) including 1 and 21,000.000,000 , are divisible by \(2,3,5,\) or 7.

Short answer

There are 16,360,000,000 numbers divisible by 2, 3, 5, or 7.

Step-by-step solution

Understand the Task

We need to determine how many numbers between 1 and 21,000,000,000 are divisible by at least one of the given numbers 2, 3, 5, or 7.

Apply the Principle of Inclusion-Exclusion (PIE)

To find the count of numbers divisible by at least one of the integers, we utilize the Inclusion-Exclusion Principle: Let \( N \) be the total count of numbers, \( |A_i| \) be the count divisible by individual numbers (2, 3, 5, 7), then:\[ |A_1 \cup A_2 \cup A_3 \cup A_4| = |A_1| + |A_2| + |A_3| + |A_4| - |A_1 \cap A_2| - |A_1 \cap A_3| - \ldots + |A_1 \cap A_2 \cap A_3| + \ldots - |A_1 \cap A_2 \cap A_3 \cap A_4| \]

Calculate Individual Divisors

Calculate the count of numbers divisible by each number individually: - Numbers divisible by 2 are given by \( \left\lfloor \frac{21000000000}{2} \right\rfloor = 10500000000 \).- Numbers divisible by 3 are \( \left\lfloor \frac{21000000000}{3} \right\rfloor = 7000000000 \).- Numbers divisible by 5 are \( \left\lfloor \frac{21000000000}{5} \right\rfloor = 4200000000 \).- Numbers divisible by 7 are \( \left\lfloor \frac{21000000000}{7} \right\rfloor = 3000000000 \).

Calculate for Pairwise Divisibility

Calculate the number of numbers divisible by the product of each pair:- Numbers divisible by 6 (2 and 3) are \( \left\lfloor \frac{21000000000}{6} \right\rfloor = 3500000000 \).- Numbers divisible by 10 (2 and 5) are \( \left\lfloor \frac{21000000000}{10} \right\rfloor = 2100000000 \).- Numbers divisible by 14 (2 and 7) are \( \left\lfloor \frac{21000000000}{14} \right\rfloor = 1500000000 \).- Numbers divisible by 15 (3 and 5) are \( \left\lfloor \frac{21000000000}{15} \right\rfloor = 1400000000 \).- Numbers divisible by 21 (3 and 7) are \( \left\lfloor \frac{21000000000}{21} \right\rfloor = 1000000000 \).- Numbers divisible by 35 (5 and 7) are \( \left\lfloor \frac{21000000000}{35} \right\rfloor = 600000000 \).

Calculate Triple Divisibility

Compute the count of numbers divisible by products of three of the numbers:- Numbers divisible by 30 (2, 3, 5) are \( \left\lfloor \frac{21000000000}{30} \right\rfloor = 700000000 \).- Numbers divisible by 42 (2, 3, 7) are \( \left\lfloor \frac{21000000000}{42} \right\rfloor = 500000000 \).- Numbers divisible by 70 (2, 5, 7) are \( \left\lfloor \frac{21000000000}{70} \right\rfloor = 300000000 \).- Numbers divisible by 105 (3, 5, 7) are \( \left\lfloor \frac{21000000000}{105} \right\rfloor = 200000000 \).

Calculate Divisibility by All Four

Determine the count of numbers divisible by 2, 3, 5, and 7: - Those divisible by 210 are \( \left\lfloor \frac{21000000000}{210} \right\rfloor = 100000000 \).

Apply Inclusion-Exclusion Formula

Substitute these values into the Inclusion-Exclusion formula:\[ |A_1 \cup A_2 \cup A_3 \cup A_4| = 10500000000 + 7000000000 + 4200000000 + 3000000000 \]\[ - (3500000000 + 2100000000 + 1500000000 + 1400000000 + 1000000000 + 600000000) \]\[ + (700000000 + 500000000 + 300000000 + 200000000) \]\[ - 100000000 \]Performing these calculations gives:\[ 24760000000 - 10100000000 + 1700000000 - 100000000 = 16360000000 \]

Verify and Finalize the Count

Ensure calculations are correct and conclude that the total count of numbers divisible by 2, 3, 5, or 7 is \( 16360000000 \).

Key concepts

Divisibility Rules

Divisibility rules are essential shortcuts in mathematics. They help us determine if a number can be divided evenly by another, without leaving a remainder. These rules are crucial in problem-solving for checking factors or multiples efficiently. For example, a number is divisible by 2 if its last digit is even. Similarly, a number is divisible by 3 if the sum of its digits is a multiple of 3.
Understanding and applying these rules simplifies the task of checking divisibility, especially with large numbers or in complex problems involving multiple divisors like 2, 3, 5, and 7. They form the foundation for more advanced mathematical concepts, such as the Inclusion-Exclusion Principle used in solving this exercise.

• Last digit even implies divisibility by 2.
• Sum of digits a multiple of 3 indicates divisibility by 3.
• A number ending in 0 or 5 shows divisibility by 5.
• For 7, double the last digit and subtract it from the rest, if the result is a multiple of 7, the original number is as well.

Mathematics Problem Solving

Mathematics problem solving involves strategizing and applying various mathematical concepts to find a solution. It includes understanding the problem statement, devising a plan, carrying out the plan, and then reviewing the solution. This approach requires logical thinking and flexibility to adapt strategies as needed.
In this exercise, problem-solving started with understanding the requirement to find how many numbers are divisible by 2, 3, 5, or 7. The plan involved using divisibility rules and the Inclusion-Exclusion Principle. Calculating step by step ensures each part of the problem is addressed, leading gradually to the final answer.

• Identify all possible solutions.
• Break the problem into manageable parts.
• Use shortcuts like divisibility rules to simplify calculations.
• Regularly check and verify your work to maintain accuracy.

Discrete Mathematics Concepts

Discrete mathematics deals with distinct and separate values or objects. Unlike calculus or algebra that involves continuous systems, discrete maths involves structures such as integers, graphs, and statements in logic. It's hugely applicable in computer science, cryptography, and combinatorics.
In this problem, we use core concepts from discrete maths such as set theory and combinatorics. Particularly, we utilize the Inclusion-Exclusion Principle for counting. This principle helps to accurately count the number of elements that meet certain conditions without overcounting any of them, which is pivotal when dealing with overlapped sets of numbers.

• Focuses on distinct values, such as whole numbers.
• Involves combinatorics, counting ways to arrange or choose items.
• Uses logic and set theory; pivotal in principles of counting.

Principles of Counting

The principles of counting govern how we determine the number of ways a given event can occur. The basic principle involves understanding permutations (order matters) and combinations (order does not matter). Counting accurately is crucial in probability, combinatory analysis, and problem-solving.
The Principle of Inclusion-Exclusion (PIE), used in this exercise, is one of these critical principles. It accounts for overlaps in sets, like multiple numbers being divisible by more than one factor. By adding and subtracting elements, PIE ensures each element is counted uniquely. Understanding these concepts allows for systematic and error-free counting.

• Comprehend different counting methods: permutations vs. combinations.
• Apply Inclusion-Exclusion to adjust for overlapping sets.
• Manipulate numbers methodically to avoid errors.