Chapter 1: Problem 13
In the above question total numbers in the set of numbers \(S=\left\\{200,201_{i \ldots, 800\\} \text { which are either divisible by } 5 \text { or }}\right.\) by 7 is: (a) 210 (b) 190 (c) 199 (d) cân't be determined
Short Answer
Expert verified
Answer: 190
Step by step solution
01
Find the number of numbers divisible by 5
First, let's determine how many numbers in the given set S are divisible by 5. To do this, we can find the least and the greatest multiple of 5 in the range of numbers from 200 to 800. Using the equation n = 5k, where n is a multiple of 5 and k is an integer, we can find the least and the greatest multiple:
Least multiple (n): 200 = 5k -> k = 40,
so 200 is the least multiple of 5 in the given set.
Greatest multiple (n): 800 = 5k -> k = 160,
so 800 is the greatest multiple of 5 in the given set.
Now, we can calculate the total number of numbers divisible by 5 in the set by subtracting the least multiple of 5 (k=40) from the greatest multiple of 5 (k=160) and adding 1: (160-40)+1 = 121.
02
Find the number of numbers divisible by 7
Next, we will determine how many numbers in the given set S are divisible by 7. Similarly, we will find the least and the greatest multiple of 7 in the range of numbers from 200 to 800. Using the equation n = 7k, where n is a multiple of 7 and k is an integer:
Least multiple (n): 203 = 7k -> k = 29,
so 203 is the least multiple of 7 in the given set.
Greatest multiple (n): 798 = 7k -> k = 114,
so 798 is the greatest multiple of 7 in the given set.
Now, we can calculate the total number of numbers divisible by 7 in the set by subtracting the least multiple of 7 (k=29) from the greatest multiple of 7 (k=114) and adding 1: (114-29)+1 = 86.
03
Find the number of numbers divisible by both 5 and 7
Since the number 35 is a multiple of both 5 and 7, we will find the least and the greatest multiple of 35 in the range of numbers from 200 to 800. Using the equation n = 35k, where n is a multiple of 35 and k is an integer:
Least multiple (n): 210 = 35k -> k = 6,
so 210 is the least multiple of 35 in the given set.
Greatest multiple (n): 770 = 35k -> k = 22,
so 770 is the greatest multiple of 35 in the given set.
Now, we can calculate the total number of numbers divisible by both 5 and 7 (i.e., by 35) in the set by subtracting the least multiple of 35 (k=6) from the greatest multiple of 35 (k=22) and adding 1: (22-6)+1 = 17.
04
Apply the principle of inclusion-exclusion
Using the principle of inclusion-exclusion, we can find the total number of numbers in the set that are divisible by either 5 or 7 as follows:
Total numbers = (Numbers divisible by 5) + (Numbers divisible by 7) - (Numbers divisible by both 5 and 7)
Total numbers = 121 + 86 - 17
Total numbers = 190
05
Choose the correct option
Now that we have calculated the total number of numbers divisible by either 5 or 7 in the given set, we can choose the correct option from the given choices:
(a) 210
(b) 190
(c) 199
(d) Can't be determined
The correct option is (b) 190.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Inclusion-Exclusion Principle
The Inclusion-Exclusion Principle is a powerful tool in combinatorics used to count the number of elements in a union of sets. It helps prevent double-counting when two or more criteria need to be considered. For example, if we want to find the count of numbers divisible by either 5 or 7 in a set, both of these conditions can sometimes overlap.
The principle states that to find the total count of numbers that satisfy at least one of the conditions, we calculate:
The principle states that to find the total count of numbers that satisfy at least one of the conditions, we calculate:
- The count of numbers satisfying the first condition (divisible by 5).
- The count of numbers satisfying the second condition (divisible by 7).
- Subtract the count of numbers satisfying both conditions (divisible by both 5 and 7), as they have been counted twice.
Divisibility
Divisibility is a key concept in arithmetic which checks whether one number can be divided by another without leaving a remainder. In this problem, we analyze numbers divisible by 5 or 7.
A number is divisible by 5 if its last digit is 0 or 5. Similarly, a number is divisible by 7 if, when you apply a specific rule or division, it leaves no remainder. For example:
A number is divisible by 5 if its last digit is 0 or 5. Similarly, a number is divisible by 7 if, when you apply a specific rule or division, it leaves no remainder. For example:
- To find numbers divisible by 5 in the range of 200 to 800, we solve for multiples of 5. We discover that the least is 200 and the greatest is 800, so we calculate that there are 121 such numbers.
- For divisibility by 7, we find the range between 203 and 798, concluding that there are 86 numbers.
Arithmetic Sequences
Arithmetic sequences are ordered lists of numbers where each term after the first is the sum of the previous term and a constant. These sequences are identified by their constant difference, known as the common difference.
In this exercise, the common difference is determined by the step values of the arithmetic sequence, which, in this context, are numbers divisible by intervals like 5 and 7. For example:
In this exercise, the common difference is determined by the step values of the arithmetic sequence, which, in this context, are numbers divisible by intervals like 5 and 7. For example:
- For multiples of 5, the sequence 200, 205, 210, etc., illustrates a common difference of 5.
- For multiples of 7, the sequence 203, 210, 217, etc., illustrates a common difference of 7.
Problem Solving Strategies
Effective problem-solving strategies are critical in approaching mathematical exercises. Here are general steps to tackle such problems:
- Understand the problem: Break down what is asked and identify the main objective.
- Explore known quantities: Calculate or assess initial factors, such as finding multiples or divisors.
- Apply relevant principles: Use mathematical principles such as inclusion-exclusion to account for overlaps.
- Check the solution: Verify results by reconsidering calculations and logical steps.