7.2 Number Theory: Primes, Factors, Multiples & Counting

Overview & Exam Relevance

Competency 002 asks you to apply number theory, including prime factorization, greatest common divisor, and divisibility rules, to whole numbers, integers, and rational numbers in problem situations. It also asks you to apply counting techniques, from forward, backward, and skip counting to combinations. Number theory explains why divisibility rules work and when a word problem calls for a greatest common factor or a least common multiple, a distinction the exam tests often.

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Elementary Number Theory

Primes, Composites, and Special Numbers

• Prime Number: A natural number strictly greater than 1 that has exactly two distinct positive divisors: 1 and itself (e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, 29).
• The Number 2: The smallest prime number, and the only even prime number.
• Composite Number: A natural number strictly greater than 1 that has more than two distinct positive divisors (e.g., 4, 6, 8, 9, 10, 12).
• The Special Cases of 0 and 1:
  • The number 1 is neither prime nor composite because it has only one positive factor (itself). Defining 1 as a prime would destroy the unique factorization property of arithmetic.
  • The number 0 is neither prime nor composite because it is not a positive natural number and has an infinite number of divisors ($0 \div n = 0$ for all $n \neq 0$).

The Fundamental Theorem of Arithmetic & Prime Factorization

Every integer strictly greater than 1 is either a prime number itself or can be represented as a unique product of prime numbers, up to the order of the factors. Prime factorizations are found using factor trees or repeated division (ladder method):

• For $360$:

$$360 = 36 \times 10 = (6 \times 6) \times (2 \times 5) = (2 \times 3 \times 2 \times 3) \times (2 \times 5) = 2^3 \times 3^2 \times 5$$

Divisibility Rules & Mathematical Justifications

Understanding the mathematical logic behind divisibility rules allows teachers to connect number theory with base-ten structure:

• Divisible by 2: The ones digit is even ($0, 2, 4, 6, 8$). Justification: Any multi-digit number can be written as $10n + d$. Because 10 is divisible by 2, $10n$ is always even, so divisibility depends entirely on the last digit $d$.
• Divisible by 3: The sum of all digits is divisible by 3. Justification: Every power of 10 is 1 greater than a multiple of 9 ($10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1$). Since 9 is divisible by 3, each power of 10 leaves a remainder of 1 times the digit, meaning the whole number leaves the same remainder as the sum of its digits.
• Divisible by 4: The number formed by the last two digits is divisible by 4. Justification: 100 is divisible by 4 ($100 = 4 \times 25$). Thus, all hundreds, thousands, and higher places are automatically divisible by 4, leaving only the tens and ones digits to consider.
• Divisible by 5: The ones digit is 0 or 5.
• Divisible by 6: The number is simultaneously divisible by both 2 (even) and 3 (digit sum divisible by 3).
• Divisible by 8: The number formed by the last three digits is divisible by 8. Justification: 1,000 is divisible by 8 ($1,000 = 8 \times 125$), so all thousands and higher places are divisible by 8.
• Divisible by 9: The sum of all digits is divisible by 9 (by the same mathematical logic as 3, since 9, 99, 999 are multiples of 9).
• Divisible by 10: The ones digit is 0.

Greatest Common Factor (GCF) versus Least Common Multiple (LCM)

Distinguishing between GCF and LCM in real-world contexts is one of the most frequently tested concepts on the TExES exam.

• Greatest Common Factor (GCF): The largest positive integer that divides two or more numbers without a remainder. Also known as the Greatest Common Divisor (GCD).
  • Prime Factorization Method: Express each number in prime-power form. The GCF is the product of all shared common prime factors, each raised to its lowest power.
  • Real-World Application: Look for scenarios involving partitioning, cutting into equal pieces, or packaging items into identical groupings with nothing left over (e.g., dividing 48 pencils and 36 notebooks into identical care packages).
• Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more numbers.
  • Prime Factorization Method: Express each number in prime-power form. The LCM is the product of every prime factor present, each raised to its highest power.
  • Real-World Application: Look for scenarios involving repeating cycles, synchronizing recurring events, or finding when two patterns will coincide again in the future (e.g., two blinking lights flashing every 8 and 12 seconds; two buses arriving at the same stop).

$$\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b$$

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Counting Strategies: From Skip Counting to Combinations

• Early counting: Counting forward and backward from any number, counting on ("8 . . . 9, 10, 11" to add 3), and skip counting by 2s, 5s, 10s, and 100s. Skip counting builds the foundation for multiplication (skip counting by 4s gives the multiples of 4).
• Organized lists and tables: Systematic listing prevents missed or repeated outcomes, for example all the two-digit numbers that can be made from the digits 1, 3, and 5 without repeating a digit (13, 15, 31, 35, 51, 53).
• Combinations: When order does not matter, students can list pairs systematically. Choosing 2 toppings from 4 (A, B, C, D) gives AB, AC, AD, BC, BD, CD: 6 combinations. Teachers connect this to the formula in the probability section.
• Using models: Counters, arrays, and tree diagrams make counting visible before formulas are introduced.
• Number theory in rational numbers: The GCF is used to simplify fractions ($\frac{18}{24} = \frac{3}{4}$ because the GCF of 18 and 24 is 6). The LCM finds a common denominator ($\frac{1}{6} + \frac{3}{8}$ uses the LCM 24: $\frac{4}{24} + \frac{9}{24} = \frac{13}{24}$).

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Classroom Scenario Application

Classroom Context: In a 4th-grade math class, Ms. Kowalski presents the following two real-world word problems during a unit on number theory:

• Problem A: "Marcus is preparing snack bags for a camping trip. He has 48 granola bars and 36 fruit snacks. What is the greatest number of identical snack bags he can make so that every bag contains the exact same contents and no snacks are left over? How many of each snack will be in a bag?"
• Problem B: "At an amusement park, the roller coaster departs every 12 minutes, and the water ride departs every 15 minutes. If both rides depart simultaneously at 10:00 AM, at what time will they next depart at the exact same moment?"

Diagnostic Analysis & Solution Path:

1. Analyzing Problem A (GCF): The problem requires partitioning items into equal groups with no remainder. This requires finding the Greatest Common Factor of 48 and 36.
   • Prime factorization of $48$: $48 = 16 \times 3 = 2^4 \times 3^1$.
   • Prime factorization of $36$: $36 = 4 \times 9 = 2^2 \times 3^2$.
   • Taking the lowest power of shared primes: $\text{GCF} = 2^2 \times 3^1 = 4 \times 3 = 12$.
   • Answer: Marcus can assemble at most 12 identical snack bags. Each bag will contain $48 \div 12 = 4$ granola bars and $36 \div 12 = 3$ fruit snacks.
2. Analyzing Problem B (LCM): The problem requires finding when two periodic cycles coincide. This requires finding the Least Common Multiple of 12 and 15.
   • Prime factorization of $12$: $12 = 2^2 \times 3^1$.
   • Prime factorization of $15$: $15 = 3^1 \times 5^1$.
   • Taking the highest power of all primes: $\text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60$.
   • Answer: The two rides will synchronize every 60 minutes (1 hour). Therefore, they will next depart together at 11:00 AM.