Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

Video Summary

The problem involves finding the greatest common divisor (GCD) of numbers of the form \( n^3(n+1)^3(n+2)^3(n+3)^3(n+4)^3 \) for nonzero natural numbers \( n \). Each term represents five consecutive integers, ensuring divisibility by at least \( 2^3 \) (due to two numbers being multiples of two, including one multiple of four), \( 3 \), and \( 5 \). Cubing each calculates the final GCD as \( 2^9 \times 3^3 \times 5^3 \). Thus, the greatest common divisor of these numbers is the product of these prime powers, which corresponds to choice E.

Keywords

greatest common divisor

GCD

consecutive integers

prime powers

natural numbers