4.5: Factors and GCF
- Page ID
- 163830
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Factors are the fundamental building blocks of numbers. They can be likened to the ingredients in a recipe, each contributing to the whole. A factor is defined as a number that divides evenly into another number, leaving no remainder.
Given two integers \(a\) and \(b\), \(a\) is said to be a factor (or divisor) of \(b\) if there exists an integer \(k\) such that \(b = a \times k\).
In other words, a factor is an integer that divides another integer without leaving a remainder.
Factors can be positive or negative, but for most elementary purposes, we typically consider only positive factors. For a positive integer \(n\), its factors are the positive integers that divide \(n\) without a remainder.
For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Each of these numbers divides into 12 without leaving a remainder.
To illustrate this concept, educators may use relatable examples such as dividing a pizza into equal slices. This tangible demonstration helps students visualize how factors work in practice.
Understanding factors is crucial as they form the basis for many mathematical concepts, including:
- Prime factorization
- Greatest Common Factor (GCF)
- Least Common Multiple (LCM)
- Divisibility rules
- Simplification of fractions
Finding Factors
The process of finding factors can be approached systematically:
- Begin with 1 and the number itself, as these are always factors.
- Check all numbers between 1 and the number.
- If division results in a whole number (no remainder), it is a factor.
Visual aids such as number lines or multiplication tables can be valuable tools in helping students recognize patterns in factors. Encouraging students to work in pairs to find factors of various numbers can promote discussion and deeper understanding of the concept.
Find the all of the factors of 24.
- Start with 1 and 24:
- 1 is always a factor (24 ÷ 1 = 24, no remainder)
- 24 is always a factor of itself (24 ÷ 24 = 1, no remainder)
- Check numbers between 1 and 24:
- 2: 24 ÷ 2 = 12 (no remainder, so 2 is a factor)
- 3: 24 ÷ 3 = 8 (no remainder, so 3 is a factor)
- 4: 24 ÷ 4 = 6 (no remainder, so 4 is a factor)
- 5: 24 ÷ 5 = 4 remainder 4 (not a factor)
- 6: 24 ÷ 6 = 4 (no remainder, so 6 is a factor)
- 7: 24 ÷ 7 = 3 remainder 3 (not a factor)
- 8: 24 ÷ 8 = 3 (no remainder, so 8 is a factor)
- 9: 24 ÷ 9 = 2 remainder 6 (not a factor)
- 10: 24 ÷ 10 = 2 remainder 4 (not a factor)
- 11: 24 ÷ 11 = 2 remainder 2 (not a factor)
- 12: 24 ÷ 12 = 2 (no remainder, so 12 is a factor)
- Numbers 13 through 23 will all leave remainders, so we can stop here.
- Compile the list of factors: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Observations:
- Pairing: Notice that the factors come in pairs. If \(a\) is a factor of 24, then 24 ÷ \(a\) is also a factor:
- 1 and 24
- 2 and 12
- 3 and 8
- 4 and 6
- Efficiency: We only needed to check up to the square root of 24 (which is approximately 4.9) to find all unique factors. After finding 4 as a factor, we automatically knew 6 was also a factor (24 ÷ 4 = 6).
- Even number: Since 24 is even, 2 is always a factor. This can be a helpful starting point for even numbers.
- Divisibility rules: Knowledge of divisibility rules can speed up the process. For example, 24 is divisible by 3 (sum of digits 2+4 = 6, which is divisible by 3) and by 4 (last two digits, 24, form a number divisible by 4).
Visual aids such as factor trees or rectangular arrays can help represent the factors of a number, especially for visual learners. Encouraging students to work in pairs to find factors of various numbers can promote discussion and a deeper understanding of the concept.
Greatest Common Factor (GCF)
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides each of the numbers without a remainder.
Formally, for two or more integers \(a, b, c, ...,\) not all zero, the GCF is the largest positive integer \(d\) such that \(d\) is a factor of each of the integers \(a, b, c, ...\)
Properties of GCF
- The \(GCF\) of any number and zero is the absolute value of the number: \(GCF(a, 0) = |a|\)
- The \(GCF\) of any two numbers is always positive, even if the numbers are negative.
- The \(GCF\) of two prime numbers is always \(1\), unless the numbers are the same prime.
- For any integers \(a\) and \(b\): \(GCF(a, b) = GCF(|a|, |b|)\)
Methods to Find GCF
There are several methods to find the \(GCF\). We'll explore three common approaches:
1. Listing Factors
This method involves listing all factors of each number and identifying the largest common factor.
Find the \(GCF\) of 48 and 72
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The common factors are: 1, 2, 3, 4, 6, 8, 12, 24 The greatest among these is 24.
Therefore, \(GCF(48, 72) = 24\)
2. Prime Factorization
This method involves breaking down each number into its prime factors and multiplying the common factors.
Example: Find the \(GCF\) of 48 and 72
\(48 = 2^4 * 3 = 2 * 2 * 2 * 2 * 3 \\ 72 = 2^3 * 3^2 = 2 * 2 * 2 * 3 * 3\)
Common factors: \(2 * 2 * 2 * 3 = 2 ^ 3 * 3 = 8 * 3 = 24\)
Therefore, \(GCF(48, 72) = 24\)
Applications of GCF
- Simplifying Fractions: To simplify a fraction, divide both the numerator and denominator by their GCF. Example: Simplify 48/72 GCF(48, 72) = 24 48/72 = (48 ÷ 24) / (72 ÷ 24) = 2/3
- Algebraic Simplification: GCF is used to factor algebraic expressions. Example: Factor 24x^2 + 36x GCF = 12x 24x^2 + 36x = 12x(2x + 3)
- Problem Solving: GCF is useful in various real-world scenarios. Example: A carpenter has planks of wood 48 inches and 72 inches long. What is the longest piece that can be cut from each plank with no waste? Solution: GCF(48, 72) = 24 inches