Mathematics
Factors
In mathematics, factors are numbers that can be multiplied together to produce a given number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Understanding factors is important in various mathematical operations, such as finding the greatest common factor or simplifying fractions.
Written by Perlego with AI-assistance
4 Key excerpts on "Factors"
Learn about this page
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
For example, let’s try to find the Factors of 12. We can find these by dividing. If you divide 12 by 12, the quotient is 1; therefore, both 12 and 1 are Factors. If you divide 12 by 6, the quotient is 2; therefore, both 6 and 2 are Factors. If you divide 12 by 4, the quotient is 3; therefore, both 4 and 3 are Factors. Conversely, if you divide 12 by 5, the quotient is 2.4—12 is not evenly divisible by 5, and, therefore, 5 is not a factor. The Factors of 12 are 1, 2, 3, 4, 6, and 12.One and the number you are trying to factor are always Factors of the number. All the other Factors will be greater than one and less than the number. It always seems easier when you pull out the factor with its partner or as a pair. A factor pair is the two numbers that are multiplied together to get the product.For example, let’s look at 18. We know 1 and 18 are Factors. Moving to 2: 18 divided by 2 is 9; therefore, 2 and 9 make up a factor pair. Moving to 3: 18 divided by 3 is 6; therefore, 3 and 6 make up a factor pair. Moving to 4: 18 is not divisible by 4. Moving to 5: 18 is not divisible by 5. Moving to 6: 6 is already on the list; you can stop. You have found all the Factors. As another example, let’s look at 50. We know that 1 and 50 are Factors. This will start our list, and we will keep going until we hit a repeat.Looking at exponentsDEFINITIONS
BaseA number that is raised to an exponent.EXAMPLE:ExponentA number that tells how many times the base is used as a factor; in an expression of the form ba , a is called the exponent, b is the base, and ba is a power of b.Exponents can be a little tricky. An exponent is like shorthand or code. You need to learn the code to decipher the meaning. You can break exponents down into four groups (we are excluding zero as a base because zero has its own rules).
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
This chapter has three main objectives. The first is to help you understand what you are doing when you perform calculations, regardless of whether you use pencil and paper, a calculator, or a computer. The second is to show that algebra greatly aids us in describing the relations between numbers and, by extension, physical quantities. The third is to show you how to tackle algebraic problems.2.1 Arithmetic with fractionsAny number we can write down, punch into a calculator, or type into a spreadsheet on a computer is a rational number; that is, it can be written as a fraction. Any integer is a fraction, n = n/1, and furthermore any decimal number that can be displayed is a fraction, for example a number like 3.14 can be regarded as a shorthand for 314/100. Arithmetic has been completely mastered when we can perform all the basic operations with fractions. And once we can do arithmetic with fractions, we can handle algebraic fractions. Practice with both the arithmetic and the algebra of fractions is provided in the End of Chapter Questions.The fundamental fraction is the reciprocal of an integerThe key to understanding fractions is first to understand what is meant by the reciprocal of an integer. The reciprocal, 1/n, of any integer, n, is defined by the fact that if we add up n of them, or equivalently if we multiply 1/n by n, we are back to 1 whole. It is the mathematical equivalent of pie slices (hence pie charts as in Chapter 9 ). If we slice a pie into 21 pieces, each is 1/21st of a pie. Once we have the reciprocal, fractions follow immediately, they are just integers multiplied by reciprocals; that is, if a and b are integers the general fraction can be written as
The reciprocal of a product is the product of the reciprocalsa b≡ a ×1 b≡1 b× a .(EQ2.1) To take an example suppose we want 21 slices. We can get to our 21 slices in at least three ways. We could just set about slicing 21 pieces. We could first cut the pie into 7 equal slices and then divide each of these into 3. Finally, we could first cut 3 slices and then divide each of these into 7. In other words if we define multiplication of reciprocals as successively subdividing into pieces then 1/21 = (1/3) × (1/7) = (1/7) × (1/3); that is, 1/(3 × 7) = (1/3) × (1/7) and we have the reciprocal of a product is the product of the reciprocals and furthermore we can multiply reciprocals in any order. Showing this formally using symbols and algebra is good practice in algebraic techniques – see End of Chapter Question 18. If we let a and b
- Alfred Posamentier, Stephen Krulik(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
Ideal recreations are those that appear difficult and yet are surprisingly simple to resolve. Such solutions can often require thinking “outside the box.” Illustrations can be found in many forms of mathematical thinking and problem processing. In selecting these motivational devices, teachers should be mindful not to make them so easy for the intended population that they become silly, or so difficult that they are beyond the reach of most students. In assessing the appropriateness of these recreational motivational devices, one needs to consider that a student’s emotional development is also influenced by Factors that occur within the classroom and can change with the various stimuli provided by the teacher. Thus teachers’ assessments of students should be flexible to anticipate changes throughout the course. As a student develops his or her intellectual capacity, so, too, will the appreciation, understanding, and processing of a puzzle evolve with time.Unfortunately too many adults still harbor a fear of or dislike of mathematics largely because their teachers did not take into account the importance of motivating instruction—especially using recreational techniques. This sort of motivational device is an excellent way of not only introducing the lesson, but also demonstrating a light-hearted aspect of mathematics that can have a lasting effect on students.Reasoning problems and mathematical games can demonstrate the fun that mathematics can offer. Such activities often win over the uninitiated and serve as much more than mere motivation for the forthcoming lesson.Topic: Identifying Factors of Numbers
An appropriate medium to display the mathematics.Materials or Equipment NeededImplementation of the Motivation Strategy
Begin the lesson by asking students to find all the proper Factors of 220 (except for 220 itself) and then finding the sum of these Factors. They should then do the same with the number 284. If they did this correctly, they should have come up with an unusual result. These two numbers can be considered “friendly numbers”!What could possibly make two numbers friendly? Your students’ first reaction might be that these numbers are friendly to them. Remind them that we are talking here about numbers that are “friendly” to each other. Mathematicians have decided that two numbers are considered friendly (or, as often used in the more sophisticated literature, “amicable”) if the sum of the proper divisors of the first number equals the second and the sum of the proper divisors of the second number equals the first.Sounds complicated? Have your students now consider the smallest pair of friendly numbers: 220 and 284. They should have gotten the following, which should show the recreational aspect of mathematics.
eBook - ePub- Alan J. Cann(Author)
- 2013(Publication Date)
- Wiley(Publisher)
5. To factor an integer, break the integer down into a group of numbers whose product equals the original number. Factors are separated by multiplication signs. Note that the number 1 is the factor of every number. All Factors of a number can be divided exactly into that number.2.6. Lowest common multiple and greatest common factor
When manipulating fractions you frequently need to find these two terms – you will see examples of this below. To find the lowest common multiple (LCM) of two numbers, make a table of multiples (e.g. 2 and 3; see Table 2.2 ). To find the greatest common factor (GCF), meaning numbers or expressions by which a larger number can be divided exactly (‘factoring’), make a table of Factors (e.g. 8 and 12; see Table 2.3 ).Table 2.2 Table of multiplesMultiples of 2 Common multiples Multiples of 3 246– ––6– 3–69 Table 2.3 Table of FactorsFactors of 8 Common Factors Factors of 12 12–48– 12–4–– 1234612 2.7. Adding and subtracting fractions
To be able to add or subtract fractions to or from fractions, the denominators must be the same (‘common’): cannot be added, but can.To find a common denominator so you can add or subtract fractions, find the LCM of all the denominators involved. Then, make the denominators equal the LCM by multiplying both the denominator and numerator by the corresponding factor of the LCM. Whenever you manipulate fractions, the final step is to reduce the answer to the lowest terms:1. Factor the numerator. 2. Factor the denominator. 3. Find the fraction mix that equals 1.Example
AddThe LCM of 3 and 5 (the denominators) is 15. Both denominators must equal the LCM, so multiply 3 by 5, and 5 by 3. Now both denominators are the same (‘common’). To avoid altering the problem, multiply the numerators by the same factor as their respective denominators. This is the same as multiplying each fraction by 15/15, i.e. by 1. Now the denominators the same, add the fractions together. You cannot reduce this fraction further, i.e. reduce numerator and denominator to their LCFs, so this is the final answer.
