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Chapter 3: Problem 67

Simplify. $$ \text { List all the factors of } 42 . \quad[2.1 \mathrm{a}] $$

Short Answer

Expert verified
Factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42.

Step by step solution

01

Understand Factors

Factors are numbers that divide another number without leaving a remainder. For example, if 3 divides 12 evenly, then 3 is a factor of 12.
02

Start with 1 and the Number Itself

Every number has 1 and itself as factors. Therefore, write down 1 and 42 as the starting factors of 42.
03

Find Pair Factors

Find other pairs of numbers that multiply together to give 42. These pairs can be found by trying to divide 42 by integers starting from 2 upwards.
04

Divide by 2

42 divided by 2 is 21, so 2 and 21 are factors.
05

Divide by 3

42 divided by 3 is 14, so 3 and 14 are factors.
06

Divide by 6

42 divided by 6 is 7, so 6 and 7 are factors.
07

Compile All Factors

List down all unique factors found: 1, 2, 3, 6, 7, 14, 21, 42.

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Key Concepts

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

Finding Factors
Finding the factors of a number means identifying all numbers that can divide the given number without leaving a remainder. A factor pairs evenly with another number to produce the original number. For instance, to find the factors of 42, we start with the basics:
  • 1 and 42 (since every number is divisible by 1 and itself)
  • Then, 2 (since 42 is even)
  • By continuing to test, we find other factors: 21, 3, 14, 6, and 7
Hence, the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.
Number Theory
Number theory involves the study of integers and their properties. One crucial aspect of this field is understanding how numbers relate through factors. Factors help us understand divisibility and the structure of numbers.
The process of finding factors relates closely to prime factorization. Breaking a number down into prime factors can further help in recognizing all factors. For example, 42 breaks down into prime factors of 2, 3, and 7. These combine to show all factors:
  • 2 x 21 = 42
  • 3 x 14 = 42
  • 6 x 7 = 42
Thus, number theory provides tools to find and utilize factors effectively.
Division
Division is the fundamental process of determining how one number can be evenly split into another. When finding factors, dividing the number by integers helps us check if it splits evenly.
Example: Dividing 42 by 2 gives 21, meaning both 2 and 21 are factors.
Steps to identify factors via division:
  • Start with 1 (every number is divisible by 1)
  • Check subsequent integers to see if they divide the number evenly
  • Use results to compile pairs of factors
Practicing division is key to mastering factor identification, and it reinforces the fundamental skill of splitting numbers accurately.

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