In this video you will learn to define factor and greatest common factor. To find all factors of a whole number. To find the GCF of two or more whole numbers. To define multiple and least common multiple. To find multiples of whole numbers. To find the LCM of two or more whole numbers. To define primes and composites. To determine if a whole number is prime or composite by examining its factors.

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## Multiplicative Relationships

In the realm of mathematics, factors play a crucial role in unlocking the mysteries of multiplicative relationships. Whether you’re a student grappling with arithmetic or a curious mind eager to delve deeper into mathematical concepts, understanding what factors are and how they work can pave the way for a solid foundation in mathematics. So, let’s embark on a journey to demystify factors and explore their significance in the world of numbers.

### What Are Factors?

Simply put, factors are the numbers that can be multiplied together to produce another number. In other words, they are the building blocks of multiplication. For example, consider the number 12. Its factors are 1, 2, 3, 4, 6, and 12, as these numbers can be multiplied in various combinations to yield 12:

1 x 12 = 12

2 x 6 = 12

3 x 4 = 12

**Basic Terminology**

Factor – A number that divides another number evenly without leaving a remainder.

Multiples – Numbers that result from multiplying a given number by an integer.

Prime Factorization – Expressing a number as the product of its prime factors.

**Significance of Factors**

Divisibility – Factors are crucial in determining the divisibility of numbers. A number is divisible by its factors without leaving a remainder. For instance, 12 is divisible by 1, 2, 3, 4, 6, and 12.

Prime Numbers – Prime numbers are numbers that have exactly two distinct factors – 1 and the number itself. Understanding factors is essential for identifying prime numbers and performing prime factorization.

Greatest Common Divisor (GCD) and Least Common Multiple (LCM) – Factors are used to find the GCD and LCM of two or more numbers. The GCD is the largest number that divides both numbers evenly, while the LCM is the smallest number that is a multiple of both numbers.

Simplifying Fractions – Factors play a crucial role in simplifying fractions. By dividing the numerator and denominator by their greatest common divisor, fractions can be expressed in their simplest form.

Real-World Applications – Factors find applications in various real-world scenarios, including:

- Calculating dimensions and factors of geometric shapes.
- Understanding patterns and relationships in data analysis.
- Solving problems in finance, engineering, and science that involve multiplicative relationships.

### What Are Multiples?

In simple terms, multiples are the product of multiplying a given number by an integer (whole number). In other words, they are the result of repeated addition of the same number. For example, consider the number 5. Its multiples are:

5 x 1 = 5

5 x 2 = 10

5 x 3 = 15

5 x 4 = 20

5 x 5 = 25

**Basic Terminology**

Multiple – A number that is the product of multiplying a given number by an integer.

Factor – A number that divides another number evenly without leaving a remainder.

Common Multiple – A number that is a multiple of two or more numbers.

Least Common Multiple (LCM) – The smallest common multiple of two or more numbers.

**Significance of Multiples**

Divisibility – Understanding multiples is crucial for determining the divisibility of numbers. A number is divisible by its multiples without leaving a remainder. For example, 15 is divisible by 3 and 5 because they are multiples of 15.

Patterns and Sequences – Multiples often form patterns and sequences that recur in various mathematical contexts. Exploring these patterns can lead to fascinating discoveries and insights into the underlying structure of numbers.

Common Multiples and LCM – Multiples are used to find common multiples and the least common multiple (LCM) of two or more numbers. The LCM is the smallest number that is a multiple of all the given numbers.

**Real-World Applications**

Multiples find applications in diverse real-world scenarios, including:

- Timekeeping and scheduling, where multiples of minutes and hours are used to organize activities.
- Finance and economics, where multiples of currency values are used to calculate profits, taxes, and investments.
- Engineering and architecture, where multiples of measurements are used to design structures and systems.

### What Are Prime Numbers?

At their core, prime numbers are the building blocks of arithmetic, the indivisible elements that cannot be formed by multiplying two smaller positive integers, other than 1 and itself. In simpler terms, a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, 11, and 13 are all prime numbers.

**Basic Terminology**

Prime Number – A natural number greater than 1 that has no positive divisors other than 1 and itself.

Composite Number – A natural number greater than 1 that is not prime.

Factorization – Expressing a number as the product of its prime factors.

Sieve of Eratosthenes – A method for finding prime numbers by eliminating multiples of known primes.

**Significance of Prime Numbers**

Fundamental Theorem of Arithmetic – Every integer greater than 1 can be uniquely expressed as a product of prime numbers, known as the prime factorization. This theorem underscores the fundamental role of prime numbers in arithmetic.

Cryptography – Prime numbers play a crucial role in modern cryptography, where they are used in encryption algorithms to secure sensitive information and communications.

Number Theory – Prime numbers are at the heart of number theory, a branch of mathematics that explores the properties and relationships of integers. They serve as the subject of numerous conjectures and unsolved problems, including the famous Riemann Hypothesis.

**Real-World Applications**

Prime numbers find applications in various real-world scenarios, including:

- Cryptography and cybersecurity, where prime numbers are used in encryption algorithms to ensure secure communication and data protection.
- Computer science and algorithms, where prime numbers are used in various computational tasks, such as hashing, searching, and optimization.
- Number theory and mathematical research, where prime numbers are studied to uncover deeper insights into the nature of mathematics and its applications.