1.3 Number Properties, Factors & Prime Factorization

1.3 Number Properties, Factors & Prime Factorization\n

A thorough understanding of number properties, prime numbers, divisibility rules, and factor relationships allows you to simplify complex arithmetic expressions quickly and accurately on the ACCUPLACER test.\n ---\n

Fundamental Properties of Whole Numbers\n

Whole numbers follow specific structural rules known as arithmetic properties. These properties allow you to rearrange and regroup terms to make mental arithmetic significantly easier.\n | Property | Addition Formula | Multiplication Formula | Key Conceptual Principle |\n| :--- | :--- | :--- | :--- |\n| Commutative | $a + b = b + a$ | $a \times b = b \times a$ | Order of terms can be swapped without changing the result. |\n| Associative | $(a + b) + c = a + (b + c)$ | $(a \times b) \times c = a \times (b \times c)$ | Grouping of terms can be changed without changing the result. |\n| Distributive | $a(b + c) = ab + ac$ | $a(b - c) = ab - ac$ | Multiplying a sum is equal to multiplying each addend individually. |\n| Identity | $a + 0 = a$ | $a \times 1 = a$ | Adding $0$ or multiplying by $1$ leaves the value unchanged. |\n| Zero Property | — | $a \times 0 = 0$ | Any number multiplied by zero equals zero. |\n

[!IMPORTANT]\n> Non-Applicability Warning: The Commutative and Associative properties apply ONLY to Addition and Multiplication. They do NOT apply to Subtraction or Division!\n> - $10 - 4 = 6$, but $4 - 10 = -6 \implies 10 - 4 \neq 4 - 10$.\n> - $24 \div 6 = 4$, but $6 \div 24 = \frac{1}{4} \implies 24 \div 6 \neq 6 \div 24$.\n ---\n

Prime vs. Composite Numbers\n

• Prime Number: A whole number strictly greater than $1$ that has exactly two distinct positive factors: $1$ and itself.\n- Composite Number: A whole number strictly greater than $1$ that has more than two positive factors.\n

Special Numbers: $0$ and $1$\n- The Number $1$ is neither prime nor composite because it has only one positive factor ($1$).\n- The Number $0$ is neither prime nor composite.\n- The Number $2$ is the only even prime number. All other even numbers are composite because they are divisible by $2$.\n

Prime Numbers under 50 Reference List\n$\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\}$\n

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Divisibility Rules\n

Divisibility rules allow you to determine whether one whole number divides into another without leaving a remainder, without performing full long division.\n | Divisor | Divisibility Condition Rule | Practical Example |\n| :--- | :--- | :--- |\n| 2 | The last digit is even ($0, 2, 4, 6, 8$). | $4,738$ ends in $8 \implies$ Divisible |\n| 3 | The sum of all digits is divisible by $3$. | $468 \to 4+6+8 = 18$ ($18 \div 3 = 6$) $\implies$ Divisible |\n| 4 | The last two digits form a number divisible by $4$. | $1,324 \to 24$ ($24 \div 4 = 6$) $\implies$ Divisible |\n| 5 | The last digit is $0$ or $5$. | $895$ ends in $5 \implies$ Divisible |\n| 6 | The number is divisible by both 2 and 3. | $468$ is even AND sum is $18 \implies$ Divisible |\n| 9 | The sum of all digits is divisible by $9$. | $7,821 \to 7+8+2+1 = 18$ ($18 \div 9 = 2$) $\implies$ Divisible |\n| 10 | The last digit is $0$. | $3,450$ ends in $0 \implies$ Divisible |\n ---\n

Prime Factorization & Factor Trees\n

The Fundamental Theorem of Arithmetic states that every integer 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.\n

Factor Tree Method for $360$\n1. Split $360$ into any two factors: $36 \times 10$.\n2. Split $36$ into $6 \times 6$; split $10$ into $2 \times 5$.\n3. Split each $6$ into $2 \times 3$.\n4. Collect all prime ends: $2, 3, 2, 3, 2, 5$.\n5. Write using exponents in ascending prime order:\n $360 = 2^3 \times 3^2 \times 5$\n

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Greatest Common Factor (GCF) & Least Common Multiple (LCM)\n

• Greatest Common Factor (GCF): The largest whole number that divides evenly into two or more given numbers without a remainder.\n- Least Common Multiple (LCM): The smallest non-zero whole number that is a multiple of two or more given numbers.\n

Method: Finding GCF and LCM via Prime Factorization\nNumbers: $72$ and $108$.\n

1. Express both numbers as prime factorizations:\n $72 = 2^3 \times 3^2$\n $108 = 2^2 \times 3^3$\n2. To find GCF: Take the smallest power of each common prime factor:\n $\text{GCF}(72, 108) = 2^2 \times 3^2 = 4 \times 9 = \mathbf{36}$\n3. To find LCM: Take the highest power of every prime factor present:\n $\text{LCM}(72, 108) = 2^3 \times 3^3 = 8 \times 27 = \mathbf{216}$\n ---\n

ACCUPLACER Exam Traps & Common Errors\n

[!CAUTION]\n> Trap 1: Confusing GCF and LCM\n> GCF produces a number less than or equal to the smallest original number. LCM produces a number greater than or equal to the largest original number. Remember: Factors are smaller (they divide into a number); Multiples are larger (a number multiplies out to them).\n [!WARNING]\n> Trap 2: Assuming Odd Numbers Are Always Prime\n> Numbers like $9, 15, 21, 25, 27, 33, 35, 39, 49, 51, 57, 91$ are odd, but they are composite because they have factors other than $1$ and themselves (e.g., $51 = 3 \times 17$, $91 = 7 \times 13$). Always test odd numbers against divisibility by $3, 5, 7$.