CHAPTER 11 – NUMBER THEORY

Reconceptualizing MathematicsPart 1: Reasoning About Numbers and Quantities

Judith Sowder, Larry Sowder, Susan Nickerson

© 2010 by W. H. Freeman and Company. All rights reserved.

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Recall the discussion of factors from Chapter 3. Since 5 6 = 30, and 15 2 = 30, each of the numbers 2, 5, 6, and 15 are factors of 30 (and of course there are more).

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Discussion

ACTIVITY

On a sheet of paper, create the array of numbers shown below:

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1. a) Cross out 1 in the array. b) The number 2 is prime. Circle 2 in the array. Cross out all the

larger multiples of 2 in the array.c) Do the same now for 3, 5, and 7.d) What is circled next? Does this procedure ever end?e) Circle 11 in the array. Cross out all the larger multiples of 11.f) Circle all the numbers not yet crossed out.g) Are the numbers circled prime? Can you surmise why it turned out this way?

2. a) If the array were extended, what column would 1000 be in? How about 1,000,000?b) 210 = 1024. What column would this be in?c) 211 = 2048. What column would this be in?

3. Find columns in the array for which the following is true: If two numbers in the column are multiplied, the product is also in that column. What is the name of this property?

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Example:

ACTIVITY

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ACTIVITY

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In a certain school there are 100 lockers lining a hallway. All are closed. Suppose 100 students walk down the hallway, in file, and the first student opens every locker. The second student comes behind the first and closes every second locker (beginning with number two). The third student changes the position of every third locker (if open, it gets closed, and vice versa). The fourth student changes the position of every fourth locker, and so on, until the 100th student changes the position of every 100th locker. After this procession, which lockers are open? Why? At the end of the procession, how many times did lockers nine and ten get changed?

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The number 6 can be written as a product of prime numbers: 2 3. The number 18 can be written as a product of three primes: 2 3 3. Can other composite numbers be written as a product of primes? These questions and others are explored next.

ACTIVITY

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ACTIVITY

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So, 2100 = 22 3 52 7

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Activity

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ACTIVITY

Return to the table we looked at in the first activity of this section. Write each number using exponents where possible. How can you determine the number of factors of a composite number by knowing the exponents of its prime factorization? Determine a rule for the number of factors for any whole number other than zero.

Hint: For any prime number to the nth power, can appear in a factor n + 1 ways where the exponents are_______ (you finish the sentence—recall the rows of factors we discussed for 72). Take note of the difference between finding factors and finding prime factors.

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It is a challenge to tackle large numbers to see whether they are prime. It was newsworthy in 1995 when a team of 600 volunteers with computers determined that a number with 129 digits was prime. It took them eight months to determine this.

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Is 495,687,115 a prime? Is 2,298,543,316 a prime? It is likely you saw immediately that five is a factor of the first number and that two is a factor of the second. Hence neither is prime.

From looking at this it is easy to see that divisibility tests for 2, 5, and 10 are quite straightforward. A divisibility test tells one whether a number is a factor of a given number without having to complete the entire division process.

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You may recall the following two properties:

1) If k is a factor of both m and n, then k is a factor of m + n.2) If k is a factor of m, but not of n, then k is not a factor of m + n.

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Determining a divisibility test for three requires more thought. The last digit is not going to be able to help us. For example, 26 is not divisible by three. And while it’s true that the prime factorization could tell us if a number is divisible by three (why?), that can be a lengthy process.

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An interesting and useful fact about dividing a number by nine is that the remainder is always the sum of digits of the number. For example, 215 ÷ 9 = 23 with a remainder of eight, and 2 + 1 + 5 = 8.

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ACTIVITY

Use the reasoning from the divisibility test for three, along with the reasoning for the remainder being the sum of digits when dividing by nine, to derive a divisibility test for 9.

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ACTIVITY

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We already know that four divides 100, 1000, 10000, and so on. So consider:

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ACTIVITY

Use the reasoning for the divisibility test for four to construct a divisibility test for eight.

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What about a divisibility test for six? Well, we already know tests for two and three. So if a number passes those two divisibility tests, then it must also be divisible by six (why?).

The same could be said for a divisibility test for 12. Here we could use the tests for four and three (how?).

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Two numbers are relatively prime if they have no prime factor in common.

Discussion

Are 12 and 6 relatively prime?

Are 15 and 6 relatively prime?Are 25 and 6 relatively prime?Are 7 and 11 relatively prime? Are any two prime numbers relatively prime?

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Divisibility tests can help us with prime factorizations of large numbers as well. Consider 12,320. We can tell right away that 2 and 5 divide 12,320, so we have:

2 5 1232We know that 4 divides 1232, so:

2 5 4 308 = 2 5 2 2 308But 4 also divides 308:

2 5 2 2 2 2 77But it is clear enough that 77 is divisible by 11:

2 2 2 2 2 5 7 11

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ACTIVITY

Use the method just explained to find the prime factorization of 1224. Try it again for 4620.

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DISCUSSION

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One important conclusion that can be made from the previous discussion is that you need to test only for divisibility by primes when deciding whether a number is prime. If, for example, 7 is not a factor, then neither will 14, 21, 28, etc. be factors.

DISCUSSION

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The greatest common factor (GCF) of two numbers m and n, sometimes also called the greatest common divisor, is the largest number that divides both m and n. In the illustration below, 6 is the GCF:

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The least common multiple (LCM) of two numbers m and n is the smallest number that is a multiple of m and also a multiple of n. In the illustration below, 48 is the LCM:

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ACTIVITY

1. What is the GCF of 68 and 102?

2. What is the LCM of 68 and 102?

3. Simplify 68/102.

4. What is 5/68 + 14/102?

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ACTIVITY

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EXAMPLE

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ACTIVITY

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ACTIVITY

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Once a number is represented as a product of prime numbers, it is quite easy to find the factors and multiples of the number. This method is possible because the prime factorization is unique. As you know, this is sometimes called the Fundamental Theorem of Arithmetic and other times is referred to as the Unique Factorization Theorem. Research has shown that students have difficulty applying this theorem.

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