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Chapter 6: Problem 20

How many positive integers between 5 and 31 a) are divisible by 3? Which integers are these? b) are divisible by 4? Which integers are these? c) are divisible by 3 and by 4? Which integers are these?

Short Answer

Expert verified
a) 9 integers: 6, 9, 12, 15, 18, 21, 24, 27, 30. b) 6 integers: 8, 12, 16, 20, 24, 28. c) 2 integers: 12, 24.

Step by step solution

01

Identify the Range

The integers we are looking at range from 6 to 30, inclusive. This ensures both bounds 5 and 31 are not included.
02

Step 1a - Identify integers divisible by 3

Find the smallest and largest integers in the range that are divisible by 3. The smallest is 6 and the largest is 30. List the integers: 6, 9, 12, 15, 18, 21, 24, 27, 30.
03

Step 2a - Count integers divisible by 3

Count the integers: There are 9 integers between 6 and 30 divisible by 3.
04

Step 1b - Identify integers divisible by 4

Find the smallest and largest integers in the range that are divisible by 4. The smallest is 8 and the largest is 28. List the integers: 8, 12, 16, 20, 24, 28.
05

Step 2b - Count integers divisible by 4

Count the integers: There are 6 integers between 6 and 30 divisible by 4.
06

Step 1c - Identify integers divisible by both 3 and 4

Find the integers that are divisible by both 3 and 4. The condition for these is that they are divisible by their least common multiple, which is 12. List the integers: 12, 24.
07

Step 2c - Count integers divisible by both 3 and 4

Count the integers: There are 2 integers between 6 and 30 divisible by both 3 and 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Integers
Positive integers are the numbers greater than zero and have no fractional or decimal part. These numbers include 1, 2, 3, and so on. They are essential when working with problems involving counting or ordering items.
In our exercise, we are focusing on positive integers within a specific range, ensuring all numbers we consider are whole and positive.
Divisibility Rules
Divisibility rules help us determine if one number can be evenly divided by another. For example, a number is divisible by 3 if the sum of its digits is divisible by 3. Similarly, a number is divisible by 4 if the last two digits form a number that is divisible by 4.
These rules are vital in identifying numbers divisible by certain integers within a given range.
Range of Numbers
The range of numbers is the spread between two bounds, such as 5 and 31. In this exercise, we primarily look at the range from 6 to 30, inclusive.
This range includes all positive integers from 6 up to and including 30, allowing us to apply divisibility rules within this subset.
Counting Integers
Counting integers is the process of finding how many numbers meet specific criteria within a given range. We often use mathematical techniques and formulas to simplify this process.
In our exercise, we count the numbers divisible by 3, 4, and both 3 and 4 within the specified range.
Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two numbers is the smallest number that is divisible by both numbers. To find the LCM, you identify the multiples of each number and then select the smallest common multiple.
For example, the LCM of 3 and 4 is 12, because 12 is the smallest number that both 3 and 4 divide into without leaving a remainder. In our exercise, the LCM helps us find numbers divisible by both 3 and 4.

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Most popular questions from this chapter

How many solutions are there to the equation $$ x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=21 $$ where \(x_{i}, i=1,2,3,4,5,\) is a nonnegative integer such that a) \(x_{1} \geq 1 ?\) b) \(x_{i} \geq 2\) for \(i=1,2,3,4,5 ?\) c) \(0 \leq x_{1} \leq 10 ?\) d) \(0 \leq x_{1} \leq 3,1 \leq x_{2}<4,\) and \(x_{3} \geq 15 ?\)

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