2.2 Divisibility, Prime Factorization, Factors, and Multiples

2.2 Divisibility, Prime Factorization, Factors, and Multiples

Number theory on the CLEP College Mathematics exam evaluates your understanding of integer structure, prime decomposition, factor/multiple relationships, and algebraic parity deduction. Mastering these core principles provides rapid shortcuts for arithmetic simplification, fraction reduction, and algebraic proof questions.

---

1. Fast Divisibility Rules & Modular Tests

Divisibility rules allow you to determine whether an integer $n$ is divisible by a divisor $d$ without executing long division.

Composite Divisibility Rule (Coprime Factorization)

To test divisibility by a composite number $C = p \cdot q$, where $p$ and $q$ are relatively prime (meaning $\text{GCD}(p, q) = 1$):

• Divisibility by 12: Must be divisible by both $3$ (sum of digits) and $4$ (last two digits).
• Divisibility by 15: Must be divisible by both $3$ (sum of digits) and $5$ (ends in $0$ or $5$).
• Divisibility by 18: Must be divisible by both $2$ (even) and $9$ (sum of digits div by $9$).

---

2. Primes, Composites & The Fundamental Theorem of Arithmetic

Definitions & Edge Cases

• Prime Number: An integer $p > 1$ that has exactly two distinct positive divisors: $1$ and $p$.
• Composite Number: An integer $n > 1$ that has more than two positive divisors (i.e., can be factored into $a \times b$ with $1 < a, b < n$).
• Special Cases:
  • The number $1$ is neither prime nor composite (it is a unit).
  • The number $0$ is neither prime nor composite.
  • The number $2$ is the smallest prime and the ONLY even prime number.

The First 25 Prime Numbers ($< 100$)

$$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$$

The Fundamental Theorem of Arithmetic

Every positive integer $n > 1$ can be uniquely factored as a product of prime powers:

$$n = p_1^{a_1} p_2^{a_2} p_3^{a_3} \cdots p_k^{a_k}$$

where $p_1 < p_2 < \dots < p_k$ are distinct prime numbers and each $a_i$ is a positive integer exponent. This factorization is unique up to the order of factors.

Formula for the Total Number of Positive Divisors

If the canonical prime factorization of $n$ is $p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, the total number of distinct positive divisors $d(n)$ is given by:

$$d(n) = (a_1 + 1)(a_2 + 1)(a_3 + 1)\cdots(a_k + 1)$$

Worked Example 1: Finding Total Divisors

Problem: Determine the total number of distinct positive divisors of $N = 360$.

1. Decompose $360$ into prime factors: $$360 = 36 \times 10 = (2^2 \cdot 3^2) \times (2 \cdot 5) = 2^3 \cdot 3^2 \cdot 5^1$$
2. Apply the divisor count formula using exponents $a_1 = 3$, $a_2 = 2$, $a_3 = 1$: $$d(360) = (3 + 1)(2 + 1)(1 + 1) = 4 \times 3 \times 2 = 24$$ Thus, $360$ has exactly $24$ positive divisors.

---

3. Greatest Common Divisor (GCD) & Least Common Multiple (LCM)

Definitions

• Greatest Common Divisor ($\text{GCD}$ or $\text{GCF}$): The largest positive integer that divides both $a$ and $b$ without remainder.
• Least Common Multiple ($\text{LCM}$): The smallest positive integer that is a multiple of both $a$ and $b$.

Prime Factorization Algorithm for GCD and LCM

Given two integers $a$ and $b$ expressed in terms of all involved prime bases:

$$a = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}, \qquad b = p_1^{b_1} p_2^{b_2} \cdots p_k^{b_k}$$

1. $\text{GCD}(a,b)$ takes the minimum exponent for each prime factor: $$\text{GCD}(a,b) = p_1^{\min(a_1,b_1)} p_2^{\min(a_2,b_2)} \cdots p_k^{\min(a_k,b_k)}$$
2. $\text{LCM}(a,b)$ takes the maximum exponent for each prime factor: $$\text{LCM}(a,b) = p_1^{\max(a_1,b_1)} p_2^{\max(a_2,b_2)} \cdots p_k^{\max(a_k,b_k)}$$

Worked Example 2: Prime Factorization Grid Method

Problem: Find the $\text{GCD}$ and $\text{LCM}$ of $72$ and $120$.

• Prime factorize each integer: $$72 = 8 \times 9 = 2^3 \cdot 3^2 \cdot 5^0$$ $$120 = 12 \times 10 = 2^3 \cdot 3^1 \cdot 5^1$$
• Extract $\text{GCD}$ (lowest exponents): $$\text{GCD}(72, 120) = 2^{\min(3,3)} \cdot 3^{\min(2,1)} \cdot 5^{\min(0,1)} = 2^3 \cdot 3^1 \cdot 5^0 = 8 \times 3 = 24$$
• Extract $\text{LCM}$ (highest exponents): $$\text{LCM}(72, 120) = 2^{\max(3,3)} \cdot 3^{\max(2,1)} \cdot 5^{\max(0,1)} = 2^3 \cdot 3^2 \cdot 5^1 = 8 \times 9 \times 5 = 360$$

The Fundamental Product Duality Theorem

For any two positive integers $a$ and $b$:

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

Verification: $24 \times 360 = 8,640$, and $72 \times 120 = 8,640$.

---

4. Parity Arithmetic & Algebraic Deduction

Parity refers to the property of an integer being either even ($2k, k \in \mathbb{Z}$) or odd ($2k + 1, k \in \mathbb{Z}$). The CLEP frequently presents abstract algebraic questions testing parity constraints.

Parity Operational Rules

+-----------------------------------------------------------------------------+
|                        PARITY ARITHMETIC RULES MATRIX                       |
|                                                                             |
|   ADDITION / SUBTRACTION               MULTIPLICATION                       |
|   Even +/- Even = Even                 Even x Even = Even                   |
|   Odd  +/- Odd  = Even                 Even x Odd  = Even                   |
|   Even +/- Odd  = Odd                  Odd  x Odd  = Odd                    |
|                                                                             |
|   EXPONENTIATION (for integer n >= 1)                                       |
|   (Even)^n = Even                      (Odd)^n = Odd                        |
+-----------------------------------------------------------------------------+

Core Deduction Principles

1. Odd Product Rule: The product of two or more integers is odd if and only if EVERY factor is odd. If even a single factor is even, the entire product is even.
2. Even Sum Rule: The sum of any number of integers is odd if and only if there is an odd number of odd terms.
3. Consecutive Integer Parity: In any set of $n$ consecutive integers, exactly one integer is divisible by $n$. For any two consecutive integers $k$ and $k+1$, exactly one is even and one is odd; their product $k(k+1)$ is always even.

Worked Example 3: Algebraic Parity Deduction

Problem: Let $m$ be an odd integer and $n$ be an even integer. Determine whether the expression $\mathcal{A} = m^2 + n^2 + 2m$ must be even or odd.

1. Evaluate $m^2$: Since $m$ is odd, $m^2 = \text{Odd} \times \text{Odd} = \text{Odd}$.
2. Evaluate $n^2$: Since $n$ is even, $n^2 = \text{Even} \times \text{Even} = \text{Even}$.
3. Evaluate $2m$: Any integer multiplied by $2$ is $\text{Even}$.
4. Sum the components: $$\mathcal{A} = \text{Odd} + \text{Even} + \text{Even} = \text{Odd}$$ Therefore, $m^2 + n^2 + 2m$ is always odd.