Number Theory 1. A unit is that by virtue of which each of the things that exist is called one. 1 A number is a multitude composed of units.

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1 Number Theory 1 The concept of number is the obvious distinction between the beast and man. Thanks to number, the cry becomes a song, noise acquires rhythm, the spring is transformed into a dance, force becomes dynamic, and outlines figures. Joseph Marie de Maistre Number Theory From the day we become aware of the world around us, we begin recognizing quantity and number. Whether it be the number of toys in our room or cereal bites on our high chair tray, we learn to count. Ancient peoples used pebbles, sticks, knots in string, tally marks in clay, then formal symbols and numeration systems to record the quantities around them. As the quantities we deal with become more complicated, we develop new numbers to record them. Our modern number system is a product of millennia of thought and theory. In this chapter, we examine the numbers we work with and what they mean. Natural Numbers A sheepherder looks out at their flocks, and notes how many sheep they have, but how to record this quantity? A set of numbers is required, and some sort of symbol to represent these numbers. Our society uses the Hindu-Arabic numerals you have seen since you were a child, with digits 0, 1,, 3, 4, 5, 6, 7, 8, 9. But what do these numbers mean? How do they behave? What operations can be performed with them, and what do the results look like? Once the sheepherder had leisure time to think, they pondered the meaning of the numbers they used. Around 300 BC, the Greek mathematician Euclid summarized the known mathematics in his work The Elements. Normally thought of as a work of geometry, The Elements also includes sections on number and number theory. Euclid defined concepts about Natural Numbers, a set with which you are very familiar with, N = {1,, 3, 4, }. Euclid called the first number, 1, the unit: A unit is that by virtue of which each of the things that exist is called one. 1 A number is a multitude composed of units. We can think of the number as composed of units, where the unit is 1, simply by recalling that =. Euclid also defined even and odd numbers using definitions that will seem very familiar to you as well, where an even number can be divided by and an odd number cannot be divided by, and differs from an even number by a unit! Surprised? Just in these first few definitions in The Elements, you can see the effect of Greek mathematics on your own mathematical education. Every natural number greater than 1 is either prime or composite. Euclid defined prime numbers as being measured only by 1, meaning the only factors of the number are 1 and itself. He defined natural numbers that were not prime as composite. Another way to define prime numbers is to state that prime numbers have only two unique factors, and thus composite numbers have more than two unique factors. With these definitions, we can Definition 1. (n.d.). Euclid's Elements, Book VII, Definitions 1 and. Retrieved June 16, 014, from Laurel Clifford Creative Commons BY-SA

2 answer the question: Is 1 a prime number? Ask yourself: how many factors does 1 have? It has only one factor, so it is not a prime number, because prime numbers have two factors. It is not composite either, as it doesn t have more than two factors. Mathematicians view the number 1 as a special number, giving it the same title Euclid did: 1 is the unit. Another way to view this unit concept is to think that the 1 item you have represents whatever units you are using to count (ounces, milligrams, feet, pickles, cats, whatever noun you are counting). The number 1 is neither prime nor composite, what about other numbers, like 57? Is 57 prime or composite? You may be pondering ideas such as: it is not an even number, so it is not divisible by ; could it be divisible by 3? How can you tell without digging out your calculator? Perhaps you know the divisibility test for determining if numbers are divisible by 3: Add the digits of the number: = 1. If the result is divisible by 3, then so is the original number: since 1 is divisible by 3, so is 57. If you grab your calculator, you can see that 57 = Arabic mathematicians of the middle ages proved divisibility tests, as did Fibonacci. The table below summarizes several divisibility tests: A number is divisible by: if the ones digit is even (divisible by ). 3 the sum of the digits is divisible by 3. 4 the last two digits form a number divisible by 4. 5 the ones digit is 0 or 5 (divisible by 5). 8 the last three digits form a number divisible by 8. 9 the sum of the digits is divisible by the ones digit is 0. The divisibility test for 7 is not given here. The work involved in determining divisibility by 7 is complicated, and arguably we re better off dividing the number by 7 to test it! What about a divisibility test for 6? Consider that 6 is product of and 3, 6 = 3, so to be divisible by 6, a number would need to pass the divisibility tests for both and 3. Look back at the table, and notice that some of the tests focus just on the last digit, while some use the sum of the digits. Our numeration system is based on sets of 10, with place values 1, 10, 100, 1000, and so forth. The number 10 is a product of and 5 (10 = 5), so any place value other than 1 is divisible by and 5. Thus the divisibility tests for and 5 only look at the digit in the ones place. The number 10 is To test divisibility by 9, each digit is added, as it represents that extra amount off from 9 in each place value. Understanding the reasoning behind the test will help you remember the test. Example 1: Divisibility Tests Use the divisibility tests to determine if 1,158,96,874,003 is a composite number. This number is too big to put in a basic 4-function calculator to divide! We can reject divisibility by, as the last digit (3) is odd. Similarly, it is not divisible by 5 or 10. To test divisibility by 3, add the digits: =54, 54 is divisible by 3, so 1,158,96,874,003 is divisible by 3 as well, and thus is a composite number (it has more than factors).

3 Number Theory 3 Try it now 1: Use the divisibility tests to determine whether, 3, 4, 5, 6, 8, 9, or 10 divide the following: a. 1,56,957,844,04 b. 3,984,670,91,570 The Greek mathematician Eratosthenes ( BC) devised a 'sieve' to discover prime numbers. A sieve is like a strainer that you drain spaghetti through when it is done cooking. The water drains out, leaving your spaghetti behind. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind. To use the sieve of Eratosthenes to find the prime numbers up to 100, make a chart of the first one hundred whole numbers (1-100): Cross out 1, because it is not prime, it is a unit. Circle, because it is the smallest positive even prime. Now cross out every multiple of. Circle 3, the next prime. Then cross out all of the multiples of 3; some multiplies of 3, like 6, may have already been crossed out because they are even. Circle the next open number, 5. Now cross out all of the multiples of 5. Circle the next open number, 7. Now cross out all of the multiples of 7. Circle any number that is left, and you have circled all the prime numbers from 1 to 100. Why didn t we have to look for multiples higher than 7? You may have noticed that the first multiple of 7 you had left to cross out was 49, and 49 = 7 7. Every multiple of 7 that was less than 49 was already crossed out because they had a smaller co-factor that was already removed. For example, 35 was already removed when we removed multiples of 5. You may also notice that 49 = 7, and 7 is the square root of 49. The largest prime that we test when looking for factors of a number will be less than or equal to the square root of the number. Try it now : Sieve of Eratosthenes Go to the website _1.html and use the applet to find the primes less than 100 via the Sieve of Eratosthenes by setting the rows to 10, and clicking on, then 3, then 5, then 7. Notice how the number of multiples removed gets smaller as the factors get larger. Sieve picture by Donovan Govan. CC-BY-SA-3.0, via Wikimedia Commons

4 4 After completing the exercise above, you should see a table listing the prime numbers less than 100: You may notice patterns and pairs of primes. Twin primes are consecutive prime numbers such as 11 and 13, and 41 and 43. It is been conjectured that there are an infinite number of twin primes, but this has never been proven. Mersenne primes are prime numbers with the form n 1, 1 less than a power of. How many Mersenne primes are there in the table? The largest prime number known to date, discovered in January 013, is a Mersenne prime, 57,885,161 1, which has 17,45,170 digits, and is also known as a titanic prime. 3 The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a unique product of prime numbers, which means that there is only one way to factor the number as primes (reordering of factors does not count as a different way). For example, the composite number 100 can be written as 5 5 or 5, but there is no other list of prime factors for 100; they all will include two 5s and two s. You may recall from previous math courses using factor trees to determine prime factors of a number. Consider the number 40. We can recognize that it is divisible by 10, and factor it as 4 10, but we haven t completed the prime factorization until we have factored the 4 and 10 into their respective prime factors: Factor trees created using the applet at: t_1.html?from=topic_t_1.html 3 Weisstein, Eric W. "Titanic Prime." From MathWorld--A Wolfram Web Resource.

5 Number Theory 5 You can also recognize that 40 is even, thus divisible by, and divide by until you reach a factor that is no longer divisible by, then divide by 3, and so forth, similar to the process used in the Sieve of Eratosthenes. This process is also known as casting out s: Both factoring methods express to us that the prime factorization is the same, 4 = ( 4 )(3)(5). Try it now 3: Find the prime factorization of the following numbers: a. 38 b. 1 c. 360 Prime factorization helps us determine the greatest common divisor (GCD), sometimes known as the greatest common factor (GCF) of two or more numbers. The greatest common divisor (GCD) is the largest natural number that divides ( goes into evenly, with no remainder) the given numbers. Since the GCD looks for common divisors, it is useful in problem solving when breaking larger amounts into smaller subsets: the GCD will be the size of the largest common subset. If we consider the numbers 40 and 360, we can find their prime factorizations express their factorizations using a Venn diagram: 40 = ( 4 )(3)(5) 360 = ( 3 )(3 )(5) 40: 360: Notice that 40 = ()()()()(3)(5) and 360 = ()()()(3)(3)(5) so they have ()()(), a 3, and a 5 in common: The intersection of the prime factorizations is the GCD: ()()()(3)(5) = 10 and is noted as GCD(40, 360) = 10. The same Venn diagram can be used to find the Least Common Multiple (LCM) of the two numbers. The Least Common Multiple (LCM) is the smallest natural number that is a 5 3 3

6 6 multiple of the given numbers (the result from multiplying each of the given numbers by a number). The LCM is useful in problem solving for predicting common repetitions of both values. The LCM can be located from the Venn diagram by listing all the factors shown in the sets, the union of the prime factorizations. The LCM of 40 and 360 would be ()()()()(3)(3)(5) = ( 4 )(3 )(5) = 70, and is noted as LCM(40, 360) = 70. The LCM represents every factor from each factorization with the highest exponent for repeated common factors. Compare the product of the LCM and GCD of 40 and 360 with the product of 40 and 360: LCM(40, 360) GCD(40, 360) = ()()()()(3)(3)(5) ()()()(3)(5) = ( 7 )(3 3 )(5 ) While = ()()()()(3)(5) ()()()(3)(3)(5) = ( 7 )(3 3 )(5 ) They both equal the same result, ( 7 )(3 3 )(5 ) = The product of the two values is the same as the product of their LCM and GCD. This relationship is helpful for checking accuracy of results as well as finding either the LCM or GCD if you know one of them. For example, if you know the GCD(40, 360) = 10, then take the product of 40 and 360, which is 86400, and divide it by 10: = 70, which is the LCM(40, 360). Try it now 4: a. Find the GCD(144, 15) and the LCM(144, 15) b. Verify the results by finding the product of and the product of the LCM GCD. Consider the values 38 and 1, and their prime factorizations: 38 = ()(19) and 1 = (3)(7). Organizing their prime factors into a Venn diagram gives us: In the intersection of the two sets, where we would normally locate the GCD, there are no values. This empty intersection tells us the GCD(38, 1) = 1, as every number has a factor of 1. When the GCD of two values is 1, we say the values are relatively prime. Notice that the LCM, the union of the sets, is the product of the factors, LCM(38, 1) = ()(19)(3)(7) = 798, which is the same as the product of 38 and 19: = : 1: When solving problems involving the LCM or GCD, we determine if we are looking for subsets (smaller sets) of the values, which would suggest a divisor or the GCD, or larger multiples (larger sets) of the values, which would suggest a multiple or the LCM. Example : Hot Dogs vs. Buns Suppose that you like a particular specialty kind of hot dog that comes in packages of 10, and you buy buns in packages of 8. How many whole packages of each should you purchase so that each hot dog has a bun?

7 Number Theory 7 Notice that we are buying whole packages, and more than one package, so we will end up with multiples of hot dogs and buns (we are not breaking packages up; grocery stores take issue with that sort of thing!). Since we don t necessarily need to feed an army, we are looking for the least common multiple: LCM(10, 8) = (5)()()() = 40 We need 40 hot dogs and 40 buns, so 4 packages of hot dogs and 5 packages of buns. The Venn diagram may seem overkill here, but did you catch how the number of packages relates to the non-common factor(s)? 10: 8: 5 Example 3: Garden Plots A large field measures 70 feet by 55 feet. If you divide it up into equal square garden plots, what size is the largest possible plot if the side lengths are natural numbers? The clue word in this problem is divide. We re dividing the larger dimensions into smaller sizes, so we are looking for the GCD. GCD(70, 55) = (7)(5) = 35 So the plots should be 35 feet by 35 feet in size. From the Venn diagram, we can see the noncommon factors tell us how many plots will fit in the field: there will be plots by 15 plots, or 30 total plots in the field. 70: 55: Try it now 5: a. Kris and Mickey are running laps around the same track. Kris can run one lap in 8 minutes but Mickey takes 1 minutes. If they both start at the same place, the same time, and run in the same direction, at what time will they first pass each other? b. What is the largest size of equal square tiles that could be used to make a checkerboard pattern on a floor measuring 18 inches by 96 inches? Beyond Natural Numbers When counting sheep, natural numbers work quite well for the sheepherder as there is no meaning to part of a sheep; sheep are whole numbers. If you re keeping track of your sheep by making a tally mark on a clay tablet, and you have no sheep, then you make no tally mark. A picture counting system needs no symbol for 0. However, when working with larger and larger numbers, making tally marks becomes cumbersome, you end up creating a positional system, as the Babylonians did, and 0 becomes important, not just to mean 0 sheep, but as a

8 8 placeholder in place value. The addition of the symbol of 0 to the natural numbers creates the set of Whole Numbers, W = {0, 1,, 3, 4, }. But will whole numbers be sufficient for all the counting and mathematical operations we need to do? An important concept for operations with number sets is the idea of closure. A set of numbers is closed under an operation if you take two numbers from the set, perform the operation, and the result is also part of the set. Consider the set of Whole Numbers and the operation of addition. If you add two whole numbers, will you always get a whole number? That is, is whole number + whole number = whole number? For example, = 8, a whole number. Hopefully you intuitively say the whole numbers are closed under the operation of addition, although we have not proven it. But are the whole numbers closed under the operation of subtraction? Is whole number whole number = whole number? For example, 3 5 = -, but wait! The result here is not a whole number, but is instead a negative number. We have a counterexample. There is no whole number that results from 3 5. If we consider a number line, showing whole numbers, view 3 5 as starting at 3 and moving left 5, there is nowhere to move to as the number line ends at 0:? Lack of closure for whole numbers suggests there s another number system out there that includes both the whole numbers and their opposites: Integers: Extending the number line to the left past 0, and using a (-) sign to show direction leftward, we can use integers to illustrate 3 5 = -, units to the left of 0. Integers are closed under the operation of subtraction. Negative numbers allow us to show distance in a direction opposite from what we call positive numbers, as we show on the left side of the number line. We use negative numbers to model debt, money is going in the opposite direction from us! The Chinese (in 00 BC) and Indians (in 60 AD) used negative numbers to model debt, although modern western society avoided their formal use until the 19 th century 4. We use negative numbers to indicate direction in temperature (below 0) as well as in altitude (below sea level). Are the integers closed under the operation of multiplication? Does an integer times an integer always produce another integer? We can try an example: (-8) 3 = -4, which is an integer. One example is not proof, but intuitively, we can argue that as multiplication can be thought of as repeated addition (add 3 sets of -8, or = -4), then as the integers are closed under addition, they should be closed under multiplication. 4 Rogers, L. (n.d.). The History of Negative Numbers. : nrich.maths.org. Retrieved June 16, 014, from

9 Number Theory 9 But what about division? Does an integer divided by an integer always produce an integer? Consider our previous example and change the operation: (-8) 3. There is no integer result here as there will be a remainder: The integers are not closed under division. So how do we split a debt of $8 among 3 people? We can have each person pay $, but that leaves debt remaining, while each person paying $3 pays too much. We need a value between $ and $3. There are no integer numbers in between consecutive integers, so we need a new number set. We need the answer to -8/3, a ratio between two integers, the Rational Numbers. Each Whole Number and Integer can be considered Rational Numbers as well, as they can be expressed as the ratio between two integers. The following are rational numbers: - as it can be written as -/1 0 as it can be written as 0/5 1.4 as it can be written as 14/10 1/ as it can be written as /4 You may notice that 1/ is already a ratio between two integers, but it could also be expressed as /4, as well as 3/6, 100/00, -5/-10, 11/, and so forth. Rational numbers do not have unique representations as a particular rational number, such as 1/, has numerous equivalent rational forms. Another property that rational numbers have: they are dense. A number set is considered to be dense if between any two numbers you can find another number that is also a member of that set. Based on this concept, are integers dense? Consider the number line: there are no integers in between any two integers. For example, there is no integer between 4 and 5. Consider the rational numbers: is there a rational number between 4/7 and 5/7? If we consider equivalent forms of 4/7 and 5/7: 4 8 and 5 10, and in between them is These forms came from multiplying the numerator and denominator by a common factor,. A student suggested that midway between 4/7 and 5/7 should be 4.5/7, using a decimal form. However, this number, 4.5/7 is not a ratio of two integers, to which the student said, Multiply the numerator and denominator by 10 to convert the decimal to an integer: , a rational number, which can be reduced to We could apply this student s technique with 4.3/7 or similar to locate another value, 43/70, in between 4/7 and 5/7.

10 10 Try it now 6: Locate a rational number between 7/15 and 8/15. Rational numbers can be expressed in decimal form. Recall that our place value system uses base 10 place values. A rational number can be viewed as a quotient: the numerator (top) divided by the denominator (bottom). Use a calculator to find the decimal forms of the following fractions and see if you can find a pattern and connection to the place value system: Every rational number in decimal form will either be a terminating (finite) or nonterminating, repeating decimal. The rational numbers that terminate have denominators (the divisor in the ratio) with only and 5 as their prime factors. If you consider that our place value system is based on multiples of 10, and prime factors of 10 are and 5, it makes sense that to terminate, the denominator needs to be a factor of a multiple of 10. A simple division process takes a rational number from fraction to decimal form. How do we go backwards from decimal to fraction form? If the decimal terminates, it is straight-forward: use the place value of the terminating digit. For example, terminates in the thousandths place, so it is 875/1000, which reduces: If the decimal does not terminate, a little bit of algebra can help. Consider the repeating decimal which can be written as 0. 8 (the bar over the value indicates that value repeats). We recognize that it is a rational number because it repeats. We know that number exists, so we call it n, n = If we multiply this number by 10 (and multiply both sides of the equation by 10), we have: 10n = Notice how this moves the decimal place one place, and there are still an infinite amount of repeating digits following. Writing the two equations together, we have: 10n = n = subtract the two (left side left side, right side right side) 9n =8 as all the repeating digits will subtract out infinitely. We now have an equation we can solve for our unknown number, n: 9n = 8 divide both sides by 9, n = 8/9 which is a ratio of two integers.

11 Number Theory 11 We can check it by using our calculator and dividing 8 by 9. Your calculator may round the last decimal place it gives you, but it should still be repeating infinitely. Example 4: Converting Repeating Decimals Convert to rational number form. We recognize that it is a rational number as it is a repeating, nonterminating decimal. We call this number n : n = We also notice that two digits are repeating, so multiply this equation by 100: 100n = , which moves the decimal place two places. 100n = n = Subtract and all the repeating decimal values will cancel out 99n = 87 Solve for n (divide by 99). n = 87/99 Check with your calculator: do Note that 87 9 since both the numerator and denominator are divisible by Example 5: Converting Repeating Decimals Convert to rational number form. We recognize that it is a rational number as it is a repeating, nonterminating decimal. We call this number n : n = We also notice that two digits are repeating (be careful as the 6 is not part of the repeating portions), so multiply this equation by 100: 100n = which moves the decimal place two places. 100n = n = Subtract and the repeating decimal values will cancel out 99n = 61.9 Solve for n (divide by 99). n = 61.9/99 WAIT! That s not done, as there s a decimal on top Note that and we now have a rational number (check it with your calculator). Converting Repeating Decimals to Rational Form: 1. Use n to represent the unknown rational form of the number.. Create a second equation by multiplying by a power of 10 based on the number of repeating digits. 3. Subtract the two equations to cancel out the repeating digits (make sure the digits align in order to do so) 4. Solve for n, reducing as necessary.

12 1 Try it now 7: Convert each decimal to rational form: a b c d Beyond Rational Numbers Ancient Greek mathematicians were very fond of rational numbers. When they discovered that there were other numbers which were not rational, they swore that "terrible" discovery to secrecy. One story (most likely just a story, but dramatically exciting anyway) suggests they murdered the man who let the secret out! Rather irrational of them. We created rational numbers to attempt to find closure under the operation of division. Are rational numbers closed under division? Is a rational number divided by a rational number always another rational number? Almost there is one number that creates havoc for division: division by 0. Rational numbers will never truly be closed under division because of division by 0. Rational numbers allowed for ratios to be expressed easily, they can't express every number. The most obvious examples can be found in geometry. Consider a square whose sides are all one unit long: Then the distance along the diagonal can be determined by the Pythagorean Theorem, a + b = c : = c = c = c We can use the square root operation to undo the squaring and solve for c: c So what does c equal? Is c a rational number? What is this number? We know that is bigger than 1, as 1 = 1. We also know it is smaller than, as = 4. So we need a rational number between 1 and. If is a rational number, then we can write it as a ratio between two integers, x and y: x y But it still has the square root in it, so let s square both sides to get rid of the square root: x y x y Which is still kind of yucky, so let s cross-multiply to get rid of the fraction: y x 1 1?

13 Number Theory 13 At this point, it still looks strange, so let s remind ourselves about that idea earlier that every number has a unique prime factorization. So whatever x is, it has a unique prime factorization. Squaring x doubles the number of prime factors. We don t know what they are, but we know there are an even number of them. Even number of factors y x Odd number of factors with Same can be said for y and y : it has an even number of prime factors. But if you look at the left side of the equation, there is a as well, an extra prime factor, so the left side has an odd number of prime factors, while the right side has an even number of prime factors. Since an even number can t equal an odd number, this situation is impossible. There is no rational number for and so has to be irrational. Challenge! 8 Apply the same argument to show that 5 is irrational and 4 is rational. Another famous irrational number (also called a transcendental number) is the ratio of the circumference of a circle to its diameter, or. It is a ratio, but cannot be expressed as the ratio of two integers. If we consider or = , we notice they are nonterminating, non-repeating decimals, so they are not rational numbers. The only way to express these numbers is to expand our number set beyond the rational number set to include numbers these new numbers, known as Irrational Numbers. The Real Number set includes both rational and irrational numbers. We can recognize irrational numbers because they will be decimals that DO NOT repeat or terminate. As we ve seen, roots are one common place that irrational numbers show up. Consider the value of the following roots: We can estimate the value of the last two roots, but cannot express that exactly using decimals. We can find roots that are perfect squares We know that 80 is close to 9, since 80 is close to 81. We know that 36 is close to 6 as 38 is close to 36.

14 14 Sometimes we can use perfect squares to simplify roots, if we recognize that the values have perfect square factors. For example, and since 16 is the perfect square of 4, we can simply it: Similarly, and since 4 is the perfect square of, we can simplify it: Is there more we could do? Does 30 have any perfect square factors? No (we re done). Real Numbers and Number Properties At this point, we ve looked at several number sets. Each number set enlarges the previous. If we drew Venn diagram 5 of the number sets, it would look like: (there s a small error on this diagram: what is labeled Natural includes 0, so it s actually the Whole number set) We ve also looked a CLOSURE, whether the result of an operation is still a member of the set. Let s revisit the closure of our number sets: Think about how the issue of closure drives the creation of new number sets. Number Set Addition Subtraction Multiplication Division Natural Closed Not Closed Closed Not Closed Numbers Whole Closed Not Closed Closed Not Closed Numbers Integers Closed Closed Closed Not Closed Rational Numbers Closed Closed Closed Closed except for division by 0 Real Numbers Closed Closed Closed Closed except for division by 0 5 Image copyright by Keith Enevoldson,

15 Number Theory 15 In addition to the closure properties, all real numbers illustrate the commutative properties of addition and multiplication, reverse the order of addition or multiplication, same result: Commutative Property: a + b = b + a ab = ba All real numbers also have the associative property of addition and multiplication, regroup the terms added or multiplied, same result: Associative Property: a + (b + c) = (a + b) + c a(bc) = (ab)c The property of real numbers that ties multiplication and addition together is the distributive property: Distributive Property: a(b + c) = ab + ac Try it now 9: We have commutative, associative and distributive properties for addition and multiplication. Do they extend to other operations? Choose values for a, b, c, etc. and test the following properties to see if they are true: a. Distributive Property for Subtraction: a(b c) = ab ac b. Distributive Property for Roots: a + b = a + b c. Distributive Property for Roots (II): a + b = a + b d. Commutative Property for Subtraction: a b = b a e. Associative Property for Subtraction: a (b c) = (a b) c

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