12.2 Number Theory, Prime Factorization, GCF/LCM, and Exponents
Key Takeaways
- Divisibility rules are mathematically justified by base-ten place value decompositions; numbers are divisible by 3 or 9 if the sum of their digits is divisible by 3 or 9, and by powers of 2 (2, 4, 8) based on their terminal $n$ digits.
- Prime numbers are integers $p > 1$ with exactly two distinct positive divisors ($1$ and $p$); $0$ and $1$ are neither prime nor composite, and $2$ is the unique even prime.
- The Fundamental Theorem of Arithmetic guarantees that every integer $n > 1$ has a unique canonical prime factorization $N = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}$, from which the total number of divisors is given by $d(N) = (a_1+1)(a_2+1)\dots(a_k+1)$.
- The Greatest Common Factor (GCF) is calculated by taking the lowest power of common prime factors, while the Least Common Multiple (LCM) takes the highest power of all appearing prime factors, satisfying $\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b$.
- Laws of exponents govern product, quotient, power-of-a-power, negative ($x^{-n} = 1/x^n$), and fractional exponent ($x^{m/n} = \sqrt[n]{x^m}$) operations, underpinning scientific notation ($a \times 10^n$, where $1 \le |a| < 10$).
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.
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
| Divisor | Divisibility Condition | Algebraic Base-10 Justification / Mathematical Proof | Worked Example |
|---|---|---|---|
| 2 | Last digit is even ($0, 2, 4, 6, 8$) | Any integer $N = 10k + d_0 = 2(5k) + d_0$. Since $2 \mid 10$, $2 \mid N \iff 2 \mid d_0$. | $4,738$: $8$ is even $\implies$ Divisible |
| 3 | Sum of the digits is divisible by $3$ | $10^n = (99\dots9) + 1$. Expanding $N = a_n10^n + \dots + a_0 = [\text{multiple of } 9] + (a_n + \dots + a_0)$. Thus $3 \mid N \iff 3 \mid \sum a_i$. | $576$: $5+7+6=18$; $3 \mid 18 \implies$ Divisible |
| 4 | The number formed by the last two digits is divisible by $4$ | $N = 100k + (10d_1 + d_0)$. Since $100 = 4 \times 25$, $4 \mid 100k$. Thus $4 \mid N \iff 4 \mid (10d_1 + d_0)$. | $15,724$: Last two digits $24$; $4 \mid 24 \implies$ Divisible |
| 5 | Last digit is $0$ or $5$ | $N = 10k + d_0 = 5(2k) + d_0$. Since $5 \mid 10$, $5 \mid N \iff d_0 \in {0, 5}$. | $8,945$: Ends in $5 \implies$ Divisible |
| 6 | Divisible by both $2$ and $3$ | Since $\gcd(2,3) = 1$, a number is divisible by $2 \times 3 = 6$ if and only if it is even and the sum of its digits is a multiple of 3. | $438$: Even and $4+3+8=15$ ($3 \mid 15$) $\implies$ Divisible |
| 8 | The number formed by the last three digits is divisible by $8$ | $N = 1,000k + (100d_2 + 10d_1 + d_0)$. Since $1,000 = 8 \times 125$, $8 \mid 1,000k$. Thus $8 \mid N \iff 8 \mid (\text{last 3 digits})$. | $35,128$: Last three digits $128$; $128 \div 8 = 16 \implies$ Divisible |
| 9 | Sum of the digits is divisible by $9$ | $N = [\text{multiple of } 9] + \sum a_i$. Since the first term is divisible by 9, $9 \mid N \iff 9 \mid \sum a_i$. | $4,896$: $4+8+9+6=27$; $9 \mid 27 \implies$ Divisible |
| 10 | Last digit is $0$ | $N = 10k + d_0$. Thus $10 \mid N \iff d_0 = 0$. | $7,390$: Ends in $0 \implies$ Divisible |
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 $N$:
- List all consecutive integers from $2$ to $N$.
- Identify the first uncrossed number ($2$) as prime; cross out all its multiples ($4, 6, 8, \dots$).
- Advance to the next uncrossed number ($3$), mark it prime, and cross out all its remaining multiples ($9, 15, 21, \dots$).
- Repeat this process for each successive prime $p$ up to $\sqrt{N}$.
- 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.
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
- Factor Tree Method: Decompose the composite number into any two factor branches and continue subdividing each branch until all terminal leaves are prime numbers.
- 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:
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 $1$:
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 $360 = 2^3 \cdot 3^2 \cdot 5^1$:
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:
- 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:
Fundamental GCF-LCM Identity
For any two positive integers $a$ and $b$:
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
| Concept | Mathematical Definition | Prime Factor Selection Rule | Key Problem Contexts & Question Phrases | Exemplar Problem |
|---|---|---|---|---|
| GCF (Greatest Common Factor) | Largest integer dividing all numbers evenly | Product of lowest powers of common primes | "Split into equal groups without leftover", "largest identical bouquets/kits", "greatest square tile size" | Partitioning $72$ pencils and $108$ markers into identical bags $\implies \text{GCF}(72, 108) = 36$ bags |
| LCM (Least Common Multiple) | Smallest integer that all numbers divide | Product of highest powers of all primes | "Repeat at the same time", "next synchronous occurrence", "minimum common length", "common denominators" | Two bells ring every $12$ min and $18$ min; when will they ring together? $\implies \text{LCM}(12, 18) = 36$ min |
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
| Exponent Law | Mathematical Identity | Conceptual Justification | Arithmetic Exemplar |
|---|---|---|---|
| Product Rule | $x^a \cdot x^b = x^{a+b}$ | $(x \dots x)_a \cdot (x \dots x)b = (x \dots x){a+b}$ | $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$ |
| Quotient Rule | $\frac{x^a}{x^b} = x^{a-b}$ ($x \ne 0$) | Canceling $b$ factors of $x$ from $a$ factors | $\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$ |
| Power of a Power | $(x^a)^b = x^{a \cdot b}$ | Multiplying $b$ groups of $x^a$ | $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$ |
| Power of a Product | $(xy)^a = x^a y^a$ | Regrouping factors via Commutative property | $(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1,000$ |
| Power of a Quotient | $\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$ ($y \ne 0$) | Multiplying fractions numerator-by-numerator | $\left(\frac{2}{3}\right)^4 = \frac{2^4}{3^4} = \frac{16}{81}$ |
| Zero Exponent | $x^0 = 1$ ($x \ne 0$) | By quotient rule: $1 = \frac{x^n}{x^n} = x^{n-n} = x^0$ | $427^0 = 1$; $(-8.5)^0 = 1$ |
| Negative Exponent | $x^{-n} = \frac{1}{x^n}$ ($x \ne 0$) | By quotient rule: $\frac{x^0}{x^n} = x^{0-n} = x^{-n} = \frac{1}{x^n}$ | $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ |
| Rational Exponent | $x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$ | $(x^{1/n})^n = x^{(1/n) \cdot n} = x^1 = x$ | $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$ |
Scientific Notation
Scientific notation expresses very large or very small real numbers compactly in the standardized form:
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:
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.
What is the total number of positive integer divisors for the composite number N = 360?
A middle school track coach needs to prepare gift bags for student athletes. The coach has 72 energy bars and 108 bottles of electrolyte drinks. If every bag must contain the exact same number of energy bars and the exact same number of electrolyte bottles with none left over, what is the MAXIMUM number of identical gift bags the coach can assemble?
Evaluate the following expression and express the final result in standard scientific notation: [(4.8 × 10⁸) × (3.0 × 10⁻³)] / (6.0 × 10²)