Precalculus Review: Real Numbers, Prime Factorization, LCM, and Fractions

Operations on Real Numbers and Algebraic Expressions

Prime and Composite Numbers

Understanding the classification of natural numbers as prime or composite is fundamental in number theory and algebra. These concepts are essential for factorization and working with fractions.

• Prime Number: A prime number is a natural number greater than 1 that is divisible only by 1 and itself.

• Composite Number: A composite number is a natural number greater than 1 that is not prime; it has divisors other than 1 and itself.

• Special Case: The number 1 is neither prime nor composite.

Examples of Prime and Composite Numbers

Number	Classification	Factors
1	Neither	1
2	Prime	1, 2
3	Prime	1, 3
4	Composite	1, 2, 4
5	Prime	1, 5
6	Composite	1, 2, 3, 6
8	Composite	1, 2, 4, 8

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. This is useful for finding greatest common divisors, least common multiples, and simplifying fractions.

• Definition: The prime factorization of a number is its expression as a product of prime numbers.

• Method: Divide the number by the smallest prime and continue dividing the quotient by primes until all factors are prime.

Example: Prime Factorization of 28

• 28 → 2 × 14

• 14 → 2 × 7

• 7 is prime

• Prime factorization: $28 = 2 \times 2 \times 7$

Example: Prime Factorization of 294

• 294 → 2 × 147

• 147 → 3 × 49

• 49 → 7 × 7

• Prime factorization: $294 = 2 \times 3 \times 7 \times 7$

Multiples and Least Common Multiple (LCM)

Multiples and the least common multiple are important for operations with fractions and solving equations involving periodicity.

• Multiple: A multiple of a number is the product of that number and a natural number.

• Least Common Multiple (LCM): The LCM of two or more natural numbers is the smallest number that is a multiple of each of the numbers.

Steps to Find the LCM

1. Write each number as a product of prime factors.

2. Identify common and remaining factors.

3. Multiply all distinct prime factors (using the highest power of each) to get the LCM.

Example: LCM of 8 and 12

• Prime factorization: $8 = 2 \times 2 \times 2$, $12 = 2 \times 2 \times 3$

• LCM: $2 \times 2 \times 2 \times 3 = 24$

Example: LCM of 18 and 15

• Prime factorization: $18 = 2 \times 3 \times 3$, $15 = 3 \times 5$

• LCM: $2 \times 3 \times 3 \times 5 = 90$

Fractions: Equivalent Fractions and Lowest Terms

Fractions are a way to represent parts of a whole. Understanding how to write equivalent fractions and reduce them to lowest terms is essential for algebraic manipulation.

• Numerator: The top number in a fraction.

• Denominator: The bottom number in a fraction.

• Equivalent Fractions: Fractions that represent the same value, even if their numerators and denominators are different.

• Lowest Terms: A fraction is in lowest terms if the numerator and denominator have no common factor other than 1.

Example: Writing Equivalent Fractions

• To write $\frac{7}{4}$ as an equivalent fraction with denominator 20:

• Find a number to multiply both numerator and denominator: $\frac{7 \times 5}{4 \times 5} = \frac{35}{20}$

Least Common Denominator (LCD)

• The LCD is the least common multiple of the denominators of a group of fractions.

• To add or compare fractions, rewrite each with the LCD as the denominator.

Example: Reducing a Fraction to Lowest Terms

• Reduce $\frac{24}{40}$ to lowest terms:

• Find the greatest common divisor (GCD) of 24 and 40, which is 8.

• Divide numerator and denominator by 8: $\frac{24 \div 8}{40 \div 8} = \frac{3}{5}$

Summary Table: Key Concepts

Concept	Definition	Example
Prime Number	Divisible only by 1 and itself	2, 3, 5, 7
Composite Number	Has divisors other than 1 and itself	4, 6, 8, 9
Prime Factorization	Product of prime numbers	$28 = 2 \times 2 \times 7$
LCM	Smallest common multiple	LCM(8,12) = 24
Equivalent Fractions	Same value, different form	$\frac{7}{4} = \frac{35}{20}$
Lowest Terms	No common factors	$\frac{24}{40} = \frac{3}{5}$

Additional info: These foundational topics are essential for success in precalculus, as they underpin more advanced concepts such as rational expressions, polynomial factorization, and solving equations.