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Chapter 5: Problem 89

List all positive integer factors of each number. $$ 18 $$

Short Answer

Expert verified
The positive integer factors of 18 are 1, 2, 3, 6, 9, and 18.

Step by step solution

01

Identify all possible factors

To find the positive integer factors of 18, start by identifying all numbers that can divide 18 without leaving a remainder. These numbers are potential factors of 18.
02

Test divisibility from 1 up to 18

Test each integer from 1 up to 18 to see if 18 is divisible by that integer without leaving a remainder. If it divides perfectly, include that number in the list of factors.
03

Determine the factors

Check each number:- 18 梅 1 = 18, remainder 0, so 1 is a factor.- 18 梅 2 = 9, remainder 0, so 2 is a factor.- 18 梅 3 = 6, remainder 0, so 3 is a factor.- 18 梅 6 = 3, remainder 0, so 6 is a factor.- 18 梅 9 = 2, remainder 0, so 9 is a factor.- 18 梅 18 = 1, remainder 0, so 18 is a factor.
04

List all positive integer factors

After checking each number, the positive integer factors of 18 are: 1, 2, 3, 6, 9, and 18.

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Key Concepts

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

Factors of Numbers
When we talk about factors of numbers, we refer to whole numbers that exactly divide another number without leaving any remainder. For instance, let's consider the number 18. We need to find all the numbers that can divide 18 completely. To do this, we systematically check every number from 1 up to 18 itself to see if it divides evenly into 18. If it does, that number is a factor. This approach helps us identify all the parts that make up the whole.
Divisibility Rules
Divisibility rules are shortcuts that help determine whether a number can be divided by another number without actually performing division. For example:
  • A number is divisible by 2 if it ends in an even number (0, 2, 4, 6, 8)
  • A number is divisible by 3 if the sum of its digits is divisible by 3
  • A number is divisible by 6 if it meets the divisibility rules for both 2 and 3
Using these rules, we can quickly find some factors. For 18, it is divisible by 2 (since it ends in 8), by 3 (since the sum of its digits 1 + 8 = 9 is divisible by 3), and by 6 because it meets both conditions.
Integer Factorization
Integer factorization involves breaking down a number into a product of smaller numbers called factors. Let's take the number 18 again. An easy way to factorize it is by using its prime factors. Prime factors of 18 are the basic building blocks, so:
18 = 2 x 3 x 3 = 2 x 3^2
This breaks 18 into its smallest parts. Each factor in this list (2, 3) is a prime number. By multiplying these prime factors in different combinations, we get all the positive integer factors of 18: 1, 2, 3, 6, 9, and 18. This method helps simplify larger numbers and understand their structure better.

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