Webinars SET B - Grade 9-10 - Sunday@6-7pm EST-Recording Webinar 3

Video Transcription

Video Summary

This lesson on number theory begins with a warm-up problem involving numbers n from 1 to 100, analyzing for how many n the number n^n is a perfect square. The key insight is that if n is even, n^n is always a perfect square; if n is odd, then n itself must be a perfect square. Counting these gives 55 numbers satisfying the condition. The lesson emphasizes the importance of parity (odd/even) in such problems.<br /><br />The instructor then introduces prime numbers and prime factorization, highlighting that every integer has a unique prime factorization useful in solving problems, especially in competitions. Concepts of greatest common divisor (GCD), least common multiple (LCM), and coprimality are discussed with examples. It covers the theorem that if a number has no prime divisors less than or equal to its square root, it is prime, reducing effort in primality tests.<br /><br />Several problems explore prime factorization in factorials, ratios of integers using LCM and GCD, and divisibility rules for numbers. Algebra is shown as a valuable tool in number theory problems, such as factorizing difference of squares, dealing with digits and place values in numbers, and solving systems with integer constraints.<br /><br />Algebraic factorization and generalized forms of numbers help solve problems involving digits repetition and prime differences. Problems on the product of digits explore possible digit compositions and their implications when adding 1.<br /><br />The lesson closes with a problem linking the product and sum of three prime numbers, demonstrating factorization and bounding through inequalities. It introduces the formula for the number of positive divisors of a number based on prime exponents in its factorization, illustrated with examples.<br /><br />A final challenging problem combines these concepts to determine divisors of multiples of N using prime exponents and factorization. The lesson concludes by encouraging understanding and practice of these tools and previews the next topic: sequences and patterns, continuing interplay between algebra and number theory.

Keywords

number theory

perfect squares

parity

prime numbers

prime factorization

greatest common divisor

least common multiple

coprimality

algebraic factorization

divisibility rules

positive divisors