Question

How many integers between 500 and 10.000 are divisible by 5 or \(7 ?\)

Short answer

2985 integers are divisible by 5 or 7 between 500 and 10,000.

Step-by-step solution

Identify Range

We are tasked with finding integers between 500 and 10,000. Therefore, our range includes all numbers from 501 to 9999.

Calculate Divisibles by 5

To find numbers divisible by 5, we take the floor of 9999 divided by 5 to get the largest integer divisible by 5, which is 1999. Similarly, for the smallest integer greater than 500 divisible by 5, we find it using the ceiling of 500 divided by 5, which gives 101. Thus, the integers divisible by 5 in this range are from 101 to 1999.

Calculate Divisibles by 7

To find numbers divisible by 7, divide 9999 by 7 and take the floor to get the largest integer divisible by 7, which is 1428. Similarly, calculate the smallest integer greater than 500 divisible by 7 using the ceiling of 500 divided by 7, resulting in 72. Therefore, integers divisible by 7 in this range are from 72 to 1428.

Calculate Common Divisibles (35)

Numbers divisible by both 5 and 7 are divisible by their least common multiple, 35. Divide 9999 by 35 and take the floor to get 285 and find the smallest integer greater than 500 divisible by 35 using the ceiling of 500 divided by 35, resulting in 15. Hence, integers divisible by 35 in this range are from 15 to 285.

Apply Inclusion-Exclusion Principle

The count of integers divisible by 5 and 7 combined is determined using the inclusion-exclusion principle: \(|A \cup B| = |A| + |B| - |A \cap B|\). From the above steps, |A| (divisible by 5) is 1999 - 100 = 1899, |B| (divisible by 7) is 1428 - 71 = 1357, |A \cap B| (divisible by 35) is 285 - 14 = 271. The total count is 1899 + 1357 - 271 = 2985.

Key concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a fundamental concept in combinatorics, used to count the number of elements in the union of multiple sets. It helps to avoid double-counting the elements that are in the intersection of these sets.To understand the Inclusion-Exclusion Principle, consider two sets, A and B. When calculating the union of A and B, we use the formula: - \(|A \cup B| = |A| + |B| - |A \cap B|\). This formula adds the count of elements in each set separately and then subtracts the number of elements common to both in order to get an accurate count. The principle can be expanded for more than two sets but the idea remains the same.In the given problem, it is applied to calculate the number of integers divisible by either 5 or 7 within a specified range by subtracting the integers counted twice – those divisible by both numbers.

Divisibility

Divisibility is an important topic in number theory concerning whether a number can be divided by another without leaving a remainder. A number, say \(n\), is divisible by another number \(d\) if \(n \div d\) results in a whole number. Understanding divisibility enables us to solve many mathematical problems efficiently.For example: - If we want to find numbers that are divisible by 5 in a certain range, we calculate their count by using the floor and ceiling functions.To solve a divisibility problem:- Determine the smallest integer in the range divisible by the desired number using the ceiling function.- Determine the largest possible integer in that range by using the floor function.For instance, calculating numbers divisible by 5 within 501 to 9999 involves finding the first multiple of 5 greater than or equal to 501 and the last one less than or equal to 9999.

Integer Number Range

When tackling problems in number theory, defining the integer number range is a crucial first step. It essentially sets the boundaries for what numbers are included in our considerations. The range dictates which elements we focus on and narrows down the potentially infinite set of numbers to just those we need. The integer number range consists of all whole numbers between two specified endpoints. For example: - A range of 500 to 10,000 implies looking at every whole number from 501 to 9999, since the boundaries themselves might not be included depending on the divisor's requirements. - Proper understanding of the range helps in accurately applying mathematical operations such as divisibility tests or counting principles within those boundaries. Clearly establishing this range simplifies the problem-solving process and ensures accuracy in any subsequent operations that rely on these values.

Problem Solving in Mathematics

Effective problem solving in mathematics extends beyond simply arriving at the right answer. It involves understanding the underlying principles and applying them systematically. Solving math problems often requires a series of clear steps based on established concepts that lead us to a logical conclusion. Consider the exercise of finding how many integers between 500 and 10,000 are divisible by 5 or 7: - First, we identify the numerical range, narrowing down our focus. - Next, we address the divisibility by 5 and 7 separately before checking divisibility by 35 (their least common multiple). Using the Inclusion-Exclusion Principle effectively prevents errors from double-counting, illustrating the importance of a methodical approach: - Breakdown complex problems into smaller, manageable parts. - Use specific mathematical tools to handle different aspects of a problem. Approaching problems this way not only aids understanding but also enhances one's ability to tackle a wide array of mathematical challenges.