12.2 Number Theory, Prime Factorization, GCF/LCM, and Exponents

12.2 Number Theory, Prime Factorization, GCF/LCM, and Exponents

CSET Focus: Number theory questions on CSET Subtest II evaluate your ability to analyze number relationships beyond rote calculation. You are expected to know the algebraic justifications behind divisibility rules, construct canonical prime power factorizations, solve real-world problem scenarios requiring GCF vs. LCM, apply exponent rules across integer and rational powers, and perform operations in scientific notation.

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1. Divisibility Rules and Mathematical Justifications

A positive integer $n$ is divisible by an integer $d$ (written $d \mid n$) if there exists an integer $k$ such that $n = d \cdot k$, meaning division yields an integer quotient with a remainder of zero.

Comprehensive Divisibility Rules & Justifications

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2. Prime and Composite Numbers

Definitions and Core Distinctions

• Prime Number: An integer $p > 1$ that has exactly two distinct positive divisors: $1$ and $p$ (e.g., $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \dots$).
• Composite Number: An integer $n > 1$ that has more than two distinct positive divisors (e.g., $4, 6, 8, 9, 10, 12, 14, 15, \dots$).
• The Numbers 0 and 1:
  • $1$ has only one positive divisor ($1$ itself). Therefore, $1$ is neither prime nor composite.
  • $0$ has infinitely many divisors ($0 = 0 \times k$ for all $k \in \mathbb{Z}^+$), but division by zero is undefined. $0$ is neither prime nor composite.
  • The Number 2: The smallest prime number and the only even prime number in existence.

The Sieve of Eratosthenes

An ancient algorithmic procedure for isolating all prime numbers up to a specified integer $N$:

1. List all consecutive integers from $2$ to $N$.
2. Identify the first uncrossed number ($2$) as prime; cross out all its multiples ($4, 6, 8, \dots$).
3. Advance to the next uncrossed number ($3$), mark it prime, and cross out all its remaining multiples ($9, 15, 21, \dots$).
4. Repeat this process for each successive prime $p$ up to $\sqrt{N}$.
5. Stopping Criterion: Any composite number $n \le N$ must possess at least one prime factor $p \le \sqrt{N}$. If an integer has no prime factors $\le \sqrt{N}$, it is guaranteed to be prime.

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3. Prime Factorization & The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic

Every integer $n > 1$ either is a prime number itself or can be represented uniquely as a product of prime numbers, up to the order of the prime factors.

Factorization Methods

1. Factor Tree Method: Decompose the composite number into any two factor branches and continue subdividing each branch until all terminal leaves are prime numbers.
2. Continuous Division (Ladder / Inverted Division) Method: Successively divide the number by the smallest possible prime divisor until the final quotient is $1$.

Canonical Prime Power Factorization Form

Every integer $N > 1$ can be expressed uniquely as:

$$N = p_1^{a_1} \cdot p_2^{a_2} \dots p_k^{a_k}$$

where $p_1 < p_2 < \dots < p_k$ are distinct prime numbers and $a_i \in \mathbb{Z}^+$ are positive integer exponents.

• Example: $360 = 2^3 \times 3^2 \times 5^1$.

Formula for the Total Number of Positive Divisors

If $N = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$, the total number of positive factors (divisors) of $N$, denoted $d(N)$, is given by the product of each exponent incremented by $1$:

$$d(N) = (a_1 + 1)(a_2 + 1)(a_3 + 1)\dots(a_k + 1)$$

Derivation: Any divisor of $N$ must have the form $p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}$, where each exponent $e_i$ can independently take any integer value from $0$ up to $a_i$ (yielding $a_i + 1$ independent choices).

• Worked Example for $360 = 2^3 \cdot 3^2 \cdot 5^1$:

$$d(360) = (3 + 1)(2 + 1)(1 + 1) = 4 \times 3 \times 2 = 24 \text{ positive divisors}.$$

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4. Greatest Common Factor (GCF) and Least Common Multiple (LCM)

Greatest Common Factor (GCF / GCD)

The Greatest Common Factor of two or more integers is the largest positive integer that divides each of the numbers without a remainder.

• Method 1 (Prime Factorization): Write the canonical prime factorizations of both numbers. The GCF is the product of the lowest power of each common prime factor:

$$\text{GCF}(a, b) = \prod p_i^{\min(a_i, b_i)}$$

• Method 2 (Euclidean Algorithm): Based on the principle that $\gcd(a, b) = \gcd(b, a \bmod b)$ where $a > b$:
  • Divide $a$ by $b$ to find quotient $q$ and remainder $r$: $a = bq + r$.
  • Replace $(a, b)$ with $(b, r)$ and repeat until remainder $r = 0$. The last non-zero remainder is $\gcd(a,b)$.

Least Common Multiple (LCM)

The Least Common Multiple of two or more integers is the smallest positive integer that is divisible by each of the given numbers.

• Method (Prime Factorization): The LCM is the product of the highest power of all prime factors occurring in either factorization:

$$\text{LCM}(a, b) = \prod p_i^{\max(a_i, b_i)}$$

Fundamental GCF-LCM Identity

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

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

Proof: For each prime $p_i$, $\min(a_i, b_i) + \max(a_i, b_i) = a_i + b_i$. Exponent addition yields $\prod p_i^{\min(a_i, b_i)} \times \prod p_i^{\max(a_i, b_i)} = \prod p_i^{a_i + b_i} = a \cdot b$.

GCF vs. LCM Problem-Solving Matrix

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5. Exponent Laws and Scientific Notation

An exponent denotes repeated multiplication: for $n \in \mathbb{Z}^+$, $x^n = \underbrace{x \cdot x \dots x}_{n \text{ times}}$, where $x$ is the base and $n$ is the power/exponent.

The Fundamental Laws of Exponents

Scientific Notation

Scientific notation expresses very large or very small real numbers compactly in the standardized form:

$$a \times 10^n$$

where $1 \le |a| < 10$ ($a$ is the coefficient/mantissa) and $n \in \mathbb{Z}$ is an integer exponent.

• Converting Standard to Scientific:
  • For numbers $\ge 10$, shift the decimal point $n$ places left ($n > 0$): $4,500,000 = 4.5 \times 10^6$.
  • For numbers $< 1$, shift the decimal point $n$ places right ($n < 0$): $0.00038 = 3.8 \times 10^{-4}$.
• Multiplication and Division in Scientific Notation:

$$(a \times 10^m) \times (b \times 10^n) = (a \cdot b) \times 10^{m+n}$$ $$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$$

Adjustment Rule: If the resulting coefficient $a \cdot b$ or $a/b$ falls outside $[1, 10)$, re-normalize the coefficient and adjust exponent $n$ accordingly.