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Questions and Answers
What is the primary focus of number theory in mathematics?
What is the primary focus of number theory in mathematics?
What is the definition of a prime number?
What is the definition of a prime number?
What is the purpose of divisibility rules in number theory?
What is the purpose of divisibility rules in number theory?
What is modular arithmetic also known as?
What is modular arithmetic also known as?
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What is the definition of a congruence in number theory?
What is the definition of a congruence in number theory?
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What is the definition of the greatest common divisor (GCD)?
What is the definition of the greatest common divisor (GCD)?
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Study Notes
Number Theory
Number theory is a branch of mathematics that deals with the properties and behavior of integers and other whole numbers.
Divisibility
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Divisibility rules: A set of rules to determine if a number is divisible by another number.
- Examples: A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. A number is divisible by 3 if the sum of its digits is divisible by 3.
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Prime numbers: Positive integers that are divisible only by themselves and 1.
- Examples: 2, 3, 5, 7, 11,...
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Composite numbers: Positive integers that are not prime.
- Examples: 4, 6, 8, 9, 10,...
Congruences
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Modular arithmetic: A system of arithmetic where numbers "wrap around" after reaching a certain value (modulus).
- Example: Clock arithmetic, where 12 + 3 = 3 (mod 12)
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Congruences: Statements that two numbers have the same remainder when divided by a third number.
- Example: 10 ≡ 2 (mod 4) because 10 and 2 have the same remainder when divided by 4.
Diophantine Equations
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Diophantine equations: Equations involving integers and polynomials, where the solutions are also integers.
- Example: 2x + 3y = 5, where x and y are integers.
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Linear Diophantine equations: Diophantine equations involving linear polynomials.
- Example: ax + by = c, where a, b, and c are integers.
Other Important Concepts
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Greatest common divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
- Example: GCD(12, 15) = 3
- Euclidean algorithm: A method for finding the GCD of two numbers.
- Fermat's Little Theorem: A theorem stating that if p is a prime number, then a^p ≡ a (mod p) for any integer a.
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Description
Test your knowledge of number theory concepts, including divisibility, prime numbers, congruences, Diophantine equations, and more. Covers fundamental principles and key formulas.
