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Understanding GCD, LCM, and Prime Numbers
Sep 2, 2024
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.
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Full transcript