Lecture Notes: Number Theory: GCD, LCM, Primes, and Composites

Overview

• The lecture focuses on number theory concepts, specifically on GCD, LCM, primes, and composites.
• Emphasis on the importance of these concepts for students in the 10th class, especially in relation to competitive exams like IOQM.

Key Concepts

Common Divisor

• A common divisor of two numbers A and B is a number that divides both A and B.
• Example: For 4 and 6, the common divisors are 1, 2.

Greatest Common Divisor (GCD)

• The GCD of A and B is the largest common divisor.
• Also known as Highest Common Factor (HCF).
• Property: If D is the GCD of A and B, then:
  • D divides A
  • D divides B

Least Common Multiple (LCM)

• The smallest positive integer that is divisible by both A and B.
• Relation between GCD and LCM:
  • Formula: GCD(A, B) * LCM(A, B) = A * B

Divisor Properties

• If a prime number P divides the product of two integers (A and B), then P must divide at least one of them.
• If P divides A^n (where n is a natural number), then P also divides A.

Prime and Composite Numbers

• Prime Number: An integer greater than 1 with exactly two distinct positive divisors: 1 and itself.
  • Examples: 2, 3, 5, 7, 11, 13...
• Composite Number: An integer greater than 1 that is not prime, i.e., it has more than two positive divisors.

Important Properties of Primes

• There exists at least one prime between any integer n and 2n.
• Every integer greater than 1 is divisible by at least one prime.
• If n is greater than 3, there exists at least one prime between n and 2n.

Factorization Example

• Expression: n^4 - 20n^2 + 4
• This expression can be factored, and the goal is to show it is not a prime number.
  • Factorization leads to two factors, indicating it cannot be prime unless one factor equals 1.

Integer n Case Analysis

• Various cases were analyzed to find integer values for n that would satisfy the expression being prime.
• Result: For all tested cases, no integer value yielded a prime result.

Conclusion

• Emphasized that for any integer n, the expression n^4 - 20n^2 + 4 is not a prime number.
• The lecture concluded with a reminder for students to review these concepts and practice related problems for better understanding.

Next Steps

• In the next class, more discussion on composite numbers and advanced prime topics is planned.