Number Theory

Definition

Number theory is the study of the properties and relationships of integers, or whole numbers. It explores fundamental questions about the nature of numbers and their behavior, focusing on concepts like divisibility, prime numbers, and patterns within the number system.

5 Must Know Facts For Your Next Test

1. Number theory is the foundation for many areas of mathematics, including cryptography, computer science, and pure mathematics.
2. The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of prime numbers.
3. Divisibility rules help determine whether a number is divisible by another number without performing long division.
4. Prime numbers are the building blocks of the integer number system and have many fascinating properties.
5. Modular arithmetic is used in computer science for tasks like hashing and generating random numbers.

Review Questions

• Explain how the concept of divisibility relates to finding multiples and factors of a number.
  • Divisibility is a key concept in number theory that directly relates to finding multiples and factors of a number. A number is divisible by another number if the first number can be evenly divided by the second without a remainder. This means that the first number is a multiple of the second number, and the second number is a factor of the first. Understanding divisibility is essential for identifying the multiples and factors of a given number, which are important operations in pre-algebra and beyond.
• Describe the role of prime numbers in the context of number theory and their connection to finding multiples and factors.
  • Prime numbers are fundamental to number theory and play a crucial role in understanding multiples and factors. A prime number is a positive integer greater than 1 that is only divisible by 1 and itself. The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of prime numbers. This means that prime numbers are the building blocks of the integer number system. When finding multiples and factors of a number, prime factorization can be used to identify the prime factors, which then determine the set of factors and multiples for that number.
• Analyze how modular arithmetic, a concept in number theory, can be applied to finding multiples and factors of numbers.
  • Modular arithmetic, a concept in number theory, can be applied to finding multiples and factors of numbers. Modular arithmetic involves performing arithmetic operations on integers, where the results are considered equivalent if they have the same remainder when divided by a fixed number, known as the modulus. This system of arithmetic can be used to identify patterns and relationships between numbers, including their multiples and factors. For example, in modular arithmetic with a modulus of 3, the multiples of 3 are the numbers that have a remainder of 0 when divided by 3. Understanding modular arithmetic can provide insights into the divisibility and factorization of numbers, which are fundamental to finding multiples and factors.

Related terms

Divisibility:

The property of a number being evenly divisible by another number, without a remainder.

Prime Numbers:

A positive integer greater than 1 that is only divisible by 1 and itself.

Modular Arithmetic: A system of arithmetic for integers, where numbers wrap around after reaching a certain value, known as the modulus.